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Currently retired.  Graduate of Electronics dept, of  Warsaw University of Technology and Mathematics dept. of Bradford University.Whole life employed in Mathematics and Information Science dept. of Warsaw University of Technology<br /><div class="js-profile-less-about u-linkUnstyled u-tcGrayDarker u-textDecorationUnderline u-displayNone">less</div></div></div><div class="ri-section"><div class="ri-section-header"><span>Interests</span></div><div class="ri-tags-container"><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="55567278" href="https://www.academia.edu/Documents/in/Characterization"><div id="js-react-on-rails-context" style="display:none" 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backbone-social-profile-documents" style="width: 100%;"><div class="u-taCenter"></div><div class="profile--tab_content_container js-tab-pane tab-pane active" id="all"><div class="profile--tab_heading_container js-section-heading" data-section="Papers" id="Papers"><h3 class="profile--tab_heading_container">Papers by Paweł J Szabłowski</h3></div><div class="js-work-strip profile--work_container" data-work-id="123611904"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/123611904/Can_the_first_two_conditional_moments_identify_a_mean_square_differentiable_process"><img alt="Research paper thumbnail of Can the first two conditional moments identify a mean square differentiable process?" class="work-thumbnail" src="https://attachments.academia-assets.com/118003768/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/123611904/Can_the_first_two_conditional_moments_identify_a_mean_square_differentiable_process">Can the first two conditional moments identify a mean square differentiable process?</a></div><div class="wp-workCard_item"><span>Computers & mathematics with applications</span><span>, 1989</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0ce0d0051bda0e5696255dd4d3360a67" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":118003768,"asset_id":123611904,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/118003768/download_file?st=MTczNDUxODQ3Myw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span 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href="https://www.academia.edu/120282242/Understanding_mathematics_of_Grover_s_algorithm"><img alt="Research paper thumbnail of Understanding mathematics of Grover’s algorithm" class="work-thumbnail" src="https://attachments.academia-assets.com/115484115/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282242/Understanding_mathematics_of_Grover_s_algorithm">Understanding mathematics of Grover’s algorithm</a></div><div class="wp-workCard_item"><span>Quantum Information Processing</span><span>, May 1, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We analyze the mathematical structure of the classical Grover's algorithm and put it within the f...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We analyze the mathematical structure of the classical Grover's algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one 'chosen' element (sometimes called a 'solution') of the dataset, but a set of m such 'chosen' elements (out of n > m). Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that 'marks,' by a suitable phase change ϕ, all these 'chosen' elements. In the first part of the paper, we construct a unique unitary operator that selects all 'chosen' elements in one step. The constructed operator is uniquely defined by the numbers ϕ and α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on α. In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, 'convenient' phase change ϕ, and by sequentially applying the so-constructed operator, we find the number of steps to find these 'chosen' elements with great probability. We apply this knowledge to study the generalizations of Grover's algorithm (m = 1, φ = π), which are of the form, the found previously, unitary operators.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a8aa8b6b117b72171216dcc6cd137617" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484115,"asset_id":120282242,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484115/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282242"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282242"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282242; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282242]").text(description); $(".js-view-count[data-work-id=120282242]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282242; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282242']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282242, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a8aa8b6b117b72171216dcc6cd137617" } } $('.js-work-strip[data-work-id=120282242]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282242,"title":"Understanding mathematics of Grover’s algorithm","translated_title":"","metadata":{"publisher":"Springer Science+Business Media","grobid_abstract":"We analyze the mathematical structure of the classical Grover's algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one 'chosen' element (sometimes called a 'solution') of the dataset, but a set of m such 'chosen' elements (out of n \u003e m). Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that 'marks,' by a suitable phase change ϕ, all these 'chosen' elements. In the first part of the paper, we construct a unique unitary operator that selects all 'chosen' elements in one step. The constructed operator is uniquely defined by the numbers ϕ and α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on α. In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, 'convenient' phase change ϕ, and by sequentially applying the so-constructed operator, we find the number of steps to find these 'chosen' elements with great probability. We apply this knowledge to study the generalizations of Grover's algorithm (m = 1, φ = π), which are of the form, the found previously, unitary operators.","publication_date":{"day":1,"month":5,"year":2021,"errors":{}},"publication_name":"Quantum Information Processing","grobid_abstract_attachment_id":115484115},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282242/Understanding_mathematics_of_Grover_s_algorithm","translated_internal_url":"","created_at":"2024-05-30T11:35:37.498-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484115,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484115/thumbnails/1.jpg","file_name":"s11128-021-03125-w.pdf","download_url":"https://www.academia.edu/attachments/115484115/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Understanding_mathematics_of_Grover_s_al.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484115/s11128-021-03125-w-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DUnderstanding_mathematics_of_Grover_s_al.pdf\u0026Expires=1734522074\u0026Signature=RnL9ogXdZGdw8f4ak1G5WlYzcZoYFq36FwZ-HeW~8d9CLPIdqIMF51NBjIFAdZdvj7GMz2Bqbzr76SzHTP4uLdeTiNrsC3gNdqKwVZHk4H4IC~PfrycqIWfWII3QM0PwZeNSfC8KXrTGNIJj4vYqaHC-mhWmLP9M2FHXMf6CA~K-VxloNJTOS-TSYJaXuPIsfqElPoWGPQgGVuT6Bt0ZEqTU1xVk715BSOAHF96yM2YgEFKRnVAoQ06UBHewnhoJRworQNEv6vahFzW8fACXUoZVnlu-~lfC3KJ4b5gOEVZ83YKs76YA3lDddM5wEuw~avX3AYCUxxGgKeHMLG38Yw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Understanding_mathematics_of_Grover_s_algorithm","translated_slug":"","page_count":21,"language":"en","content_type":"Work","summary":"We analyze the mathematical structure of the classical Grover's algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one 'chosen' element (sometimes called a 'solution') of the dataset, but a set of m such 'chosen' elements (out of n \u003e m). Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that 'marks,' by a suitable phase change ϕ, all these 'chosen' elements. In the first part of the paper, we construct a unique unitary operator that selects all 'chosen' elements in one step. The constructed operator is uniquely defined by the numbers ϕ and α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on α. In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, 'convenient' phase change ϕ, and by sequentially applying the so-constructed operator, we find the number of steps to find these 'chosen' elements with great probability. We apply this knowledge to study the generalizations of Grover's algorithm (m = 1, φ = π), which are of the form, the found previously, unitary operators.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484115,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484115/thumbnails/1.jpg","file_name":"s11128-021-03125-w.pdf","download_url":"https://www.academia.edu/attachments/115484115/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Understanding_mathematics_of_Grover_s_al.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484115/s11128-021-03125-w-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DUnderstanding_mathematics_of_Grover_s_al.pdf\u0026Expires=1734522074\u0026Signature=RnL9ogXdZGdw8f4ak1G5WlYzcZoYFq36FwZ-HeW~8d9CLPIdqIMF51NBjIFAdZdvj7GMz2Bqbzr76SzHTP4uLdeTiNrsC3gNdqKwVZHk4H4IC~PfrycqIWfWII3QM0PwZeNSfC8KXrTGNIJj4vYqaHC-mhWmLP9M2FHXMf6CA~K-VxloNJTOS-TSYJaXuPIsfqElPoWGPQgGVuT6Bt0ZEqTU1xVk715BSOAHF96yM2YgEFKRnVAoQ06UBHewnhoJRworQNEv6vahFzW8fACXUoZVnlu-~lfC3KJ4b5gOEVZ83YKs76YA3lDddM5wEuw~avX3AYCUxxGgKeHMLG38Yw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":318,"name":"Mathematical Physics","url":"https://www.academia.edu/Documents/in/Mathematical_Physics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":444,"name":"Quantum Computing","url":"https://www.academia.edu/Documents/in/Quantum_Computing"},{"id":518,"name":"Quantum Physics","url":"https://www.academia.edu/Documents/in/Quantum_Physics"},{"id":2640,"name":"Quantum Information","url":"https://www.academia.edu/Documents/in/Quantum_Information"},{"id":4199,"name":"Quantum Information Processing","url":"https://www.academia.edu/Documents/in/Quantum_Information_Processing"},{"id":26817,"name":"Algorithm","url":"https://www.academia.edu/Documents/in/Algorithm"}],"urls":[{"id":42488752,"url":"https://link.springer.com/content/pdf/10.1007/s11128-021-03125-w.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282241"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282241/Towards_a_q_analogue_of_the_Kibble_Slepian_formula_in_3_dimensions"><img alt="Research paper thumbnail of Towards a q-analogue of the Kibble–Slepian formula in 3 dimensions" class="work-thumbnail" src="https://attachments.academia-assets.com/115484123/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282241/Towards_a_q_analogue_of_the_Kibble_Slepian_formula_in_3_dimensions">Towards a q-analogue of the Kibble–Slepian formula in 3 dimensions</a></div><div class="wp-workCard_item"><span>Journal of Functional Analysis</span><span>, 2012</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The gener...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q−Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3 dimensional Normal distribution. We show that this is not the case. We indicate some values of the parameters and some compact set in R 3 of positive measure, such that the values of the extension of KS formula are on this set negative. Nevertheless we indicate other applications of so generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam-Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey-Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we indicate some new interesting properties of the Askey-Wilson polynomials with complex parameters. We also pose several open questions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f2786ecaed38008a146129a4f891087a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484123,"asset_id":120282241,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484123/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282241"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282241"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282241; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282241]").text(description); $(".js-view-count[data-work-id=120282241]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282241; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282241']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282241, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "f2786ecaed38008a146129a4f891087a" } } $('.js-work-strip[data-work-id=120282241]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282241,"title":"Towards a q-analogue of the Kibble–Slepian formula in 3 dimensions","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q−Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3 dimensional Normal distribution. We show that this is not the case. We indicate some values of the parameters and some compact set in R 3 of positive measure, such that the values of the extension of KS formula are on this set negative. Nevertheless we indicate other applications of so generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam-Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey-Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we indicate some new interesting properties of the Askey-Wilson polynomials with complex parameters. We also pose several open questions.","publication_date":{"day":null,"month":null,"year":2012,"errors":{}},"publication_name":"Journal of Functional Analysis","grobid_abstract_attachment_id":115484123},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282241/Towards_a_q_analogue_of_the_Kibble_Slepian_formula_in_3_dimensions","translated_internal_url":"","created_at":"2024-05-30T11:35:37.285-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484123,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484123/thumbnails/1.jpg","file_name":"1011.4929v5.pdf","download_url":"https://www.academia.edu/attachments/115484123/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Towards_a_q_analogue_of_the_Kibble_Slepi.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484123/1011.4929v5-libre.pdf?1717097620=\u0026response-content-disposition=attachment%3B+filename%3DTowards_a_q_analogue_of_the_Kibble_Slepi.pdf\u0026Expires=1734522074\u0026Signature=V7YwkUx8zn-QZ~3lgJtNXV6VPnjUtFPl2MTFKHtvrySYfoJ2oAZ7j5i-O1cSP31Ah6towz6oOktskmM72MvP7M85IvUAEqUdKYFQ3BAlf6mFcvE~lR4eO2yse4IFni79WgYgTyuNwXIXOC8Iz9OvoDuEmD~CCzlWBNJxNdeno3W74fVqxP0xPPZKVer0yJkYdt1sZdscWDKGS9eEZwqbzvGuEQ9dMlfi2e-~AoB4uH6UI-gxTYVzGvkxQIZC068sx0SmWrZRewhdHQAv4YjZFzm-M1MyKNrm3ZoivdFF8GJGMfY9~jsfqpPZd5UUejnj-dKd~-9sse4~wxraA98kTg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Towards_a_q_analogue_of_the_Kibble_Slepian_formula_in_3_dimensions","translated_slug":"","page_count":20,"language":"en","content_type":"Work","summary":"We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q−Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3 dimensional Normal distribution. We show that this is not the case. We indicate some values of the parameters and some compact set in R 3 of positive measure, such that the values of the extension of KS formula are on this set negative. Nevertheless we indicate other applications of so generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam-Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey-Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we indicate some new interesting properties of the Askey-Wilson polynomials with complex parameters. We also pose several open questions.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484123,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484123/thumbnails/1.jpg","file_name":"1011.4929v5.pdf","download_url":"https://www.academia.edu/attachments/115484123/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Towards_a_q_analogue_of_the_Kibble_Slepi.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484123/1011.4929v5-libre.pdf?1717097620=\u0026response-content-disposition=attachment%3B+filename%3DTowards_a_q_analogue_of_the_Kibble_Slepi.pdf\u0026Expires=1734522074\u0026Signature=V7YwkUx8zn-QZ~3lgJtNXV6VPnjUtFPl2MTFKHtvrySYfoJ2oAZ7j5i-O1cSP31Ah6towz6oOktskmM72MvP7M85IvUAEqUdKYFQ3BAlf6mFcvE~lR4eO2yse4IFni79WgYgTyuNwXIXOC8Iz9OvoDuEmD~CCzlWBNJxNdeno3W74fVqxP0xPPZKVer0yJkYdt1sZdscWDKGS9eEZwqbzvGuEQ9dMlfi2e-~AoB4uH6UI-gxTYVzGvkxQIZC068sx0SmWrZRewhdHQAv4YjZFzm-M1MyKNrm3ZoivdFF8GJGMfY9~jsfqpPZd5UUejnj-dKd~-9sse4~wxraA98kTg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":371,"name":"Functional Analysis","url":"https://www.academia.edu/Documents/in/Functional_Analysis"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":267802,"name":"Dimensional","url":"https://www.academia.edu/Documents/in/Dimensional"},{"id":346242,"name":"Hermite Polynomials","url":"https://www.academia.edu/Documents/in/Hermite_Polynomials"},{"id":555663,"name":"Mathematics and Probability","url":"https://www.academia.edu/Documents/in/Mathematics_and_Probability"},{"id":1142720,"name":"Normal Distribution","url":"https://www.academia.edu/Documents/in/Normal_Distribution"},{"id":2902615,"name":"Cartesian product","url":"https://www.academia.edu/Documents/in/Cartesian_product"}],"urls":[{"id":42488751,"url":"https://doi.org/10.1016/j.jfa.2011.09.007"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282240"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282240/On_affinity_relating_two_positive_measures_and_the_connection_coefficients_between_polynomials_orthogonalized_by_these_measures"><img alt="Research paper thumbnail of On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures" class="work-thumbnail" src="https://attachments.academia-assets.com/115484113/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282240/On_affinity_relating_two_positive_measures_and_the_connection_coefficients_between_polynomials_orthogonalized_by_these_measures">On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures</a></div><div class="wp-workCard_item"><span>Applied Mathematics and Computation</span><span>, Feb 1, 2013</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We consider two positive, normalized measures dA (x) and dB (x) related by the relationship dA (x...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We consider two positive, normalized measures dA (x) and dB (x) related by the relationship dA (x) = C x+D dB (x) or by dA (x) = C x 2 +E dB (x) and dB (x) is symmetric. We show that then the polynomial sequences {an (x)} , {bn (x)} orthogonal with respect to these measures are related by the relationship an (x) = bn (x) + κnb n−1 (x) or by an (x) = bn (x) + λnb n−2 (x) for some sequences {κn} and {λn}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {bn (x)} and the sequence {κn} that have a form of Fourier series expansion of the Radon-Nikodym derivative of one measure with respect to the other.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="08ac44f55ab683251c23920a27b6b54b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484113,"asset_id":120282240,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484113/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282240"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282240"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282240; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282240]").text(description); $(".js-view-count[data-work-id=120282240]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282240; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282240']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282240, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "08ac44f55ab683251c23920a27b6b54b" } } $('.js-work-strip[data-work-id=120282240]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282240,"title":"On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We consider two positive, normalized measures dA (x) and dB (x) related by the relationship dA (x) = C x+D dB (x) or by dA (x) = C x 2 +E dB (x) and dB (x) is symmetric. We show that then the polynomial sequences {an (x)} , {bn (x)} orthogonal with respect to these measures are related by the relationship an (x) = bn (x) + κnb n−1 (x) or by an (x) = bn (x) + λnb n−2 (x) for some sequences {κn} and {λn}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {bn (x)} and the sequence {κn} that have a form of Fourier series expansion of the Radon-Nikodym derivative of one measure with respect to the other.","publication_date":{"day":1,"month":2,"year":2013,"errors":{}},"publication_name":"Applied Mathematics and Computation","grobid_abstract_attachment_id":115484113},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282240/On_affinity_relating_two_positive_measures_and_the_connection_coefficients_between_polynomials_orthogonalized_by_these_measures","translated_internal_url":"","created_at":"2024-05-30T11:35:37.072-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484113,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484113/thumbnails/1.jpg","file_name":"1202.pdf","download_url":"https://www.academia.edu/attachments/115484113/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_affinity_relating_two_positive_measur.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484113/1202-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DOn_affinity_relating_two_positive_measur.pdf\u0026Expires=1734522074\u0026Signature=UYCTS8k4RCKZ1ewzuZjY~cnfDOmEN2c5GZ8cHjIMfuloOLInyuvX7sZlWDq-61qFbeQzollMzU3qejFHCcym792uYCwsB4mGa5XlOb64iDhvzauWb7XCl7rMfQZFZhMQKk36fsyYNPzxt1qXwYda8PGLWlzOPg2~Mv955fgqXJGF49G9pE~qltJ6kW3ACHYKGIsbmYOcfuyNLc08W9tG5MrOdwSq7lWqhte2LtzkPwFtmgUUS5xJBRqElhkMsCEciKwgChj09iP8LLm-zWvkIq~zRfnpaC7~G4k-WW~JB4z8WOPNx7O0mC6cMCjrPw4L0wArUVjRN05rllvDtIolHA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_affinity_relating_two_positive_measures_and_the_connection_coefficients_between_polynomials_orthogonalized_by_these_measures","translated_slug":"","page_count":13,"language":"en","content_type":"Work","summary":"We consider two positive, normalized measures dA (x) and dB (x) related by the relationship dA (x) = C x+D dB (x) or by dA (x) = C x 2 +E dB (x) and dB (x) is symmetric. We show that then the polynomial sequences {an (x)} , {bn (x)} orthogonal with respect to these measures are related by the relationship an (x) = bn (x) + κnb n−1 (x) or by an (x) = bn (x) + λnb n−2 (x) for some sequences {κn} and {λn}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {bn (x)} and the sequence {κn} that have a form of Fourier series expansion of the Radon-Nikodym derivative of one measure with respect to the other.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484113,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484113/thumbnails/1.jpg","file_name":"1202.pdf","download_url":"https://www.academia.edu/attachments/115484113/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_affinity_relating_two_positive_measur.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484113/1202-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DOn_affinity_relating_two_positive_measur.pdf\u0026Expires=1734522074\u0026Signature=UYCTS8k4RCKZ1ewzuZjY~cnfDOmEN2c5GZ8cHjIMfuloOLInyuvX7sZlWDq-61qFbeQzollMzU3qejFHCcym792uYCwsB4mGa5XlOb64iDhvzauWb7XCl7rMfQZFZhMQKk36fsyYNPzxt1qXwYda8PGLWlzOPg2~Mv955fgqXJGF49G9pE~qltJ6kW3ACHYKGIsbmYOcfuyNLc08W9tG5MrOdwSq7lWqhte2LtzkPwFtmgUUS5xJBRqElhkMsCEciKwgChj09iP8LLm-zWvkIq~zRfnpaC7~G4k-WW~JB4z8WOPNx7O0mC6cMCjrPw4L0wArUVjRN05rllvDtIolHA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":115484112,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484112/thumbnails/1.jpg","file_name":"1202.pdf","download_url":"https://www.academia.edu/attachments/115484112/download_file","bulk_download_file_name":"On_affinity_relating_two_positive_measur.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484112/1202-libre.pdf?1717097617=\u0026response-content-disposition=attachment%3B+filename%3DOn_affinity_relating_two_positive_measur.pdf\u0026Expires=1734522074\u0026Signature=U~QsemYQtpVbuKDfZijIRKlqk39uw0G-BuAg6XKd62R-wuOLquG-PTVcJoUl0qLTsKnXeM1nZMlmDBs9FrYA0ATHa1Rev-IU18ORROQa5YdNSghy1BBMbYpuloZuwf6g2CDTRNZ4L2ifzAQNzxxP0ehceAMdG8DUS6tsomEq4UxsHAD0KRnHxsBPMMfWjQHGOHMfjQkHmw8jyWkvU6kBD1Lp2nuRyjjaORDyl8yBAwl24A~nCxdWROsseUgzrDsl1KhinzJU3QPQ8ykkWHarJmj6Igstv1Dyc6I1~VppwCrpzmCYPXZsYGqXJxIYLi16cXXju6YI8Kr0B5-yZ2glxg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":12026,"name":"Orthogonal polynomials","url":"https://www.academia.edu/Documents/in/Orthogonal_polynomials"},{"id":556845,"name":"Numerical Analysis and Computational Mathematics","url":"https://www.academia.edu/Documents/in/Numerical_Analysis_and_Computational_Mathematics"},{"id":1228829,"name":"Fourier Series","url":"https://www.academia.edu/Documents/in/Fourier_Series"},{"id":3777178,"name":"Polynomial","url":"https://www.academia.edu/Documents/in/Polynomial"}],"urls":[{"id":42488750,"url":"http://arxiv.org/pdf/1202.1606"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282239"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282239/Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths"><img alt="Research paper thumbnail of Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths" class="work-thumbnail" src="https://attachments.academia-assets.com/115484116/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282239/Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths">Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jun 23, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We are studying stationary random processes with conditional polynomial moments that allow a cont...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification, are important because they are relatively easy to simulate. One does not have to care about the distribution of their jumps which is always difficult to find. Among those processes with the continuous path are the Ornstein-Uhlenbeck process, the Gamma process, the process with Arcsin or Wigner margins and the Theta functions as the transition densities and others. We give a simple criterion for the stationary process to have a continuous path modification expressed in terms of skewness and excess kurtosis of the marginal distribution.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3bafd370e1429d60d119eb0e2fb7bbfc" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484116,"asset_id":120282239,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484116/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282239"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282239"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282239; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282239]").text(description); $(".js-view-count[data-work-id=120282239]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282239; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282239']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282239, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "3bafd370e1429d60d119eb0e2fb7bbfc" } } $('.js-work-strip[data-work-id=120282239]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282239,"title":"Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths","translated_title":"","metadata":{"publisher":"Cornell University","ai_title_tag":"Stationary Markov Processes with Polynomial Moments","grobid_abstract":"We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification, are important because they are relatively easy to simulate. One does not have to care about the distribution of their jumps which is always difficult to find. Among those processes with the continuous path are the Ornstein-Uhlenbeck process, the Gamma process, the process with Arcsin or Wigner margins and the Theta functions as the transition densities and others. We give a simple criterion for the stationary process to have a continuous path modification expressed in terms of skewness and excess kurtosis of the marginal distribution.","publication_date":{"day":23,"month":6,"year":2022,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":115484116},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282239/Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths","translated_internal_url":"","created_at":"2024-05-30T11:35:36.848-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484116,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484116/thumbnails/1.jpg","file_name":"2206.11798.pdf","download_url":"https://www.academia.edu/attachments/115484116/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Stationary_Markov_stochastic_processes_w.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484116/2206.11798-libre.pdf?1717097620=\u0026response-content-disposition=attachment%3B+filename%3DStationary_Markov_stochastic_processes_w.pdf\u0026Expires=1734522074\u0026Signature=LrI~Mzma6NAjRDmAtPFu9GwIJYMtJHW6nP9LgmoaycWM7d8XElIQ-m0DMWwN1vz-urRC0vs8H3K5yp1fy1U4z-32ePYgpwT3IhO2e-3qdqB3u~CNm5~MYnz5hfWzH-jBd6XZvUJRUjfQi0zdq9SSFmqbRQChOcGhWMcd~OpKApXPQtx4E3E4bdixeu9KBTASKmoyEV6TQTYVOeZsY2Txb1EFZkI4pf1A2Dnel4MzfyT6tj33ucLRVZ2ewZ6ntim47hdx8SJB-zeXTJwo3LRg4qCtPxgvKhd~tcRUxaQVcO22Y8c2HVgXxEykhBDl5kiYAC8uSTjQLoBmX7T8RO6xQw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths","translated_slug":"","page_count":20,"language":"en","content_type":"Work","summary":"We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification, are important because they are relatively easy to simulate. One does not have to care about the distribution of their jumps which is always difficult to find. Among those processes with the continuous path are the Ornstein-Uhlenbeck process, the Gamma process, the process with Arcsin or Wigner margins and the Theta functions as the transition densities and others. We give a simple criterion for the stationary process to have a continuous path modification expressed in terms of skewness and excess kurtosis of the marginal distribution.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484116,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484116/thumbnails/1.jpg","file_name":"2206.11798.pdf","download_url":"https://www.academia.edu/attachments/115484116/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Stationary_Markov_stochastic_processes_w.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484116/2206.11798-libre.pdf?1717097620=\u0026response-content-disposition=attachment%3B+filename%3DStationary_Markov_stochastic_processes_w.pdf\u0026Expires=1734522074\u0026Signature=LrI~Mzma6NAjRDmAtPFu9GwIJYMtJHW6nP9LgmoaycWM7d8XElIQ-m0DMWwN1vz-urRC0vs8H3K5yp1fy1U4z-32ePYgpwT3IhO2e-3qdqB3u~CNm5~MYnz5hfWzH-jBd6XZvUJRUjfQi0zdq9SSFmqbRQChOcGhWMcd~OpKApXPQtx4E3E4bdixeu9KBTASKmoyEV6TQTYVOeZsY2Txb1EFZkI4pf1A2Dnel4MzfyT6tj33ucLRVZ2ewZ6ntim47hdx8SJB-zeXTJwo3LRg4qCtPxgvKhd~tcRUxaQVcO22Y8c2HVgXxEykhBDl5kiYAC8uSTjQLoBmX7T8RO6xQw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":16460,"name":"Statistical Physics","url":"https://www.academia.edu/Documents/in/Statistical_Physics"},{"id":191286,"name":"Kurtosis","url":"https://www.academia.edu/Documents/in/Kurtosis"},{"id":480922,"name":"Skewness","url":"https://www.academia.edu/Documents/in/Skewness"},{"id":3777178,"name":"Polynomial","url":"https://www.academia.edu/Documents/in/Polynomial"}],"urls":[{"id":42488749,"url":"https://arxiv.org/pdf/2206.11798"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282238"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282238/Some_Remarks_Concerning_the_Dual_Control_Problem"><img alt="Research paper thumbnail of Some Remarks Concerning the Dual Control Problem" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282238/Some_Remarks_Concerning_the_Dual_Control_Problem">Some Remarks Concerning the Dual Control Problem</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">ABSTRACT L&amp;#39;auteur considere le probleme simultane d&amp;#39;estimation des parametres d&a...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">ABSTRACT L&amp;#39;auteur considere le probleme simultane d&amp;#39;estimation des parametres d&amp;#39;un systeme et du controle optimal de ce meme systeme. Plus precisement on considere un systeme donne par: xsub(i+1)=f(xsubi, u subi, n, p), pinVlhkRsupd avec une fonction de cout de la forme: J(pi )=lim infsub(Narrrinfin) E(sumsub(i=0)sup(N - 1) rho (usubi, x subi)). Dans une premiere partie, l&amp;#39;auteur propose une generalisation de la martingale de Mandl (cas lineaire quadratique) et montre que pour tout suite d&amp;#39;estimateurs (psubi) de (psubi) &amp;quot;stabilisante et reguliere&amp;quot; dans un certain sens on a: limsub(Narrrinfin) (1/N)sumsub(i=0)sup(N - 1) rho (xsubi, u(xsubi, psubi))=J(pi )=infsub pi J(pi ) ou pi =(u(x sub0, psub0), u(xsub1, psub1), ...). Ces resultats sont ensuite appliques au cas lineaire quadratique et permettent de reobtenir les resultats principaux de Mandl. Ensuite est proposee une methode (appelee naturelle) d&amp;#39;estimation de la forme psub(i+1)= pi subi=psubi - mu subiFsub(i+1)(omega , psubi) si pi subiinM ou psub(i+1)= projection de pi subi sur M si pi subi notinM, ou Fsub(i+1)(omega , p) est une suite de vecteurs aleatoires p- dimensionnels, (mu subi) une suite de reels positifs avec mu sub0=1, mu subiin(0, 1) et MlhkRsupp est ferme convexe. Sous certaines conditions de regularite cette procedure d&amp;#39;estimation converge vers p, et de plus pi =(u(xsub0, psub0), u(xsub0, psub1), ...) realise infsub pi J(pi ). Ainsi le probleme d&amp;#39;estimation et de controle du systeme se remene au probleme de la convergence de cette procedure d&amp;#39;estimation. (For the entire collection see MR 80f:90006a.)</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282238"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282238"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282238; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282238]").text(description); $(".js-view-count[data-work-id=120282238]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282238; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282238']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282238, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=120282238]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282238,"title":"Some Remarks Concerning the Dual Control Problem","translated_title":"","metadata":{"abstract":"ABSTRACT L\u0026amp;#39;auteur considere le probleme simultane d\u0026amp;#39;estimation des parametres d\u0026amp;#39;un systeme et du controle optimal de ce meme systeme. Plus precisement on considere un systeme donne par: xsub(i+1)=f(xsubi, u subi, n, p), pinVlhkRsupd avec une fonction de cout de la forme: J(pi )=lim infsub(Narrrinfin) E(sumsub(i=0)sup(N - 1) rho (usubi, x subi)). Dans une premiere partie, l\u0026amp;#39;auteur propose une generalisation de la martingale de Mandl (cas lineaire quadratique) et montre que pour tout suite d\u0026amp;#39;estimateurs (psubi) de (psubi) \u0026amp;quot;stabilisante et reguliere\u0026amp;quot; dans un certain sens on a: limsub(Narrrinfin) (1/N)sumsub(i=0)sup(N - 1) rho (xsubi, u(xsubi, psubi))=J(pi )=infsub pi J(pi ) ou pi =(u(x sub0, psub0), u(xsub1, psub1), ...). Ces resultats sont ensuite appliques au cas lineaire quadratique et permettent de reobtenir les resultats principaux de Mandl. Ensuite est proposee une methode (appelee naturelle) d\u0026amp;#39;estimation de la forme psub(i+1)= pi subi=psubi - mu subiFsub(i+1)(omega , psubi) si pi subiinM ou psub(i+1)= projection de pi subi sur M si pi subi notinM, ou Fsub(i+1)(omega , p) est une suite de vecteurs aleatoires p- dimensionnels, (mu subi) une suite de reels positifs avec mu sub0=1, mu subiin(0, 1) et MlhkRsupp est ferme convexe. Sous certaines conditions de regularite cette procedure d\u0026amp;#39;estimation converge vers p, et de plus pi =(u(xsub0, psub0), u(xsub0, psub1), ...) realise infsub pi J(pi ). Ainsi le probleme d\u0026amp;#39;estimation et de controle du systeme se remene au probleme de la convergence de cette procedure d\u0026amp;#39;estimation. (For the entire collection see MR 80f:90006a.)","publication_date":{"day":17,"month":12,"year":2020,"errors":{}}},"translated_abstract":"ABSTRACT L\u0026amp;#39;auteur considere le probleme simultane d\u0026amp;#39;estimation des parametres d\u0026amp;#39;un systeme et du controle optimal de ce meme systeme. Plus precisement on considere un systeme donne par: xsub(i+1)=f(xsubi, u subi, n, p), pinVlhkRsupd avec une fonction de cout de la forme: J(pi )=lim infsub(Narrrinfin) E(sumsub(i=0)sup(N - 1) rho (usubi, x subi)). Dans une premiere partie, l\u0026amp;#39;auteur propose une generalisation de la martingale de Mandl (cas lineaire quadratique) et montre que pour tout suite d\u0026amp;#39;estimateurs (psubi) de (psubi) \u0026amp;quot;stabilisante et reguliere\u0026amp;quot; dans un certain sens on a: limsub(Narrrinfin) (1/N)sumsub(i=0)sup(N - 1) rho (xsubi, u(xsubi, psubi))=J(pi )=infsub pi J(pi ) ou pi =(u(x sub0, psub0), u(xsub1, psub1), ...). Ces resultats sont ensuite appliques au cas lineaire quadratique et permettent de reobtenir les resultats principaux de Mandl. Ensuite est proposee une methode (appelee naturelle) d\u0026amp;#39;estimation de la forme psub(i+1)= pi subi=psubi - mu subiFsub(i+1)(omega , psubi) si pi subiinM ou psub(i+1)= projection de pi subi sur M si pi subi notinM, ou Fsub(i+1)(omega , p) est une suite de vecteurs aleatoires p- dimensionnels, (mu subi) une suite de reels positifs avec mu sub0=1, mu subiin(0, 1) et MlhkRsupp est ferme convexe. Sous certaines conditions de regularite cette procedure d\u0026amp;#39;estimation converge vers p, et de plus pi =(u(xsub0, psub0), u(xsub0, psub1), ...) realise infsub pi J(pi ). Ainsi le probleme d\u0026amp;#39;estimation et de controle du systeme se remene au probleme de la convergence de cette procedure d\u0026amp;#39;estimation. (For the entire collection see MR 80f:90006a.)","internal_url":"https://www.academia.edu/120282238/Some_Remarks_Concerning_the_Dual_Control_Problem","translated_internal_url":"","created_at":"2024-05-30T11:35:36.640-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Some_Remarks_Concerning_the_Dual_Control_Problem","translated_slug":"","page_count":null,"language":"fr","content_type":"Work","summary":"ABSTRACT L\u0026amp;#39;auteur considere le probleme simultane d\u0026amp;#39;estimation des parametres d\u0026amp;#39;un systeme et du controle optimal de ce meme systeme. Plus precisement on considere un systeme donne par: xsub(i+1)=f(xsubi, u subi, n, p), pinVlhkRsupd avec une fonction de cout de la forme: J(pi )=lim infsub(Narrrinfin) E(sumsub(i=0)sup(N - 1) rho (usubi, x subi)). Dans une premiere partie, l\u0026amp;#39;auteur propose une generalisation de la martingale de Mandl (cas lineaire quadratique) et montre que pour tout suite d\u0026amp;#39;estimateurs (psubi) de (psubi) \u0026amp;quot;stabilisante et reguliere\u0026amp;quot; dans un certain sens on a: limsub(Narrrinfin) (1/N)sumsub(i=0)sup(N - 1) rho (xsubi, u(xsubi, psubi))=J(pi )=infsub pi J(pi ) ou pi =(u(x sub0, psub0), u(xsub1, psub1), ...). Ces resultats sont ensuite appliques au cas lineaire quadratique et permettent de reobtenir les resultats principaux de Mandl. Ensuite est proposee une methode (appelee naturelle) d\u0026amp;#39;estimation de la forme psub(i+1)= pi subi=psubi - mu subiFsub(i+1)(omega , psubi) si pi subiinM ou psub(i+1)= projection de pi subi sur M si pi subi notinM, ou Fsub(i+1)(omega , p) est une suite de vecteurs aleatoires p- dimensionnels, (mu subi) une suite de reels positifs avec mu sub0=1, mu subiin(0, 1) et MlhkRsupp est ferme convexe. Sous certaines conditions de regularite cette procedure d\u0026amp;#39;estimation converge vers p, et de plus pi =(u(xsub0, psub0), u(xsub0, psub1), ...) realise infsub pi J(pi ). Ainsi le probleme d\u0026amp;#39;estimation et de controle du systeme se remene au probleme de la convergence de cette procedure d\u0026amp;#39;estimation. (For the entire collection see MR 80f:90006a.)","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[],"research_interests":[{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":3273604,"name":"Dual (grammatical number)","url":"https://www.academia.edu/Documents/in/Dual_grammatical_number_"}],"urls":[{"id":42488748,"url":"https://doi.org/10.1201/9781003071877-10"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282237"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282237/On_connection_between_values_of_Riemann_zeta_function_at_rationals_and_generalized_harmonic_numbers"><img alt="Research paper thumbnail of On connection between values of Riemann zeta function at rationals and generalized harmonic numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/115484121/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282237/On_connection_between_values_of_Riemann_zeta_function_at_rationals_and_generalized_harmonic_numbers">On connection between values of Riemann zeta function at rationals and generalized harmonic numbers</a></div><div class="wp-workCard_item"><span>Mathematica Eterna</span><span>, Jan 12, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer and rational values of arguments to certain rapidly converging series, where some generalized harmonic numbers appear. Most of the results of the paper can be derived from the recent, more advanced results, on the properties of Arakawa-Kaneko zeta functions. We derive our results directly, by solving simple recursions. The form of mentioned above generalized harmonic numbers carries information, about the values of the arguments of Hurwitz function. In particular we prove: 8k 2 N : (k; 1) = (k) = 2 k 1 2 k 1 1</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fddd9654b255cce3acc47ec8705741be" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484121,"asset_id":120282237,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484121/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282237"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282237"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282237; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282237]").text(description); $(".js-view-count[data-work-id=120282237]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282237; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282237']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282237, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "fddd9654b255cce3acc47ec8705741be" } } $('.js-work-strip[data-work-id=120282237]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282237,"title":"On connection between values of Riemann zeta function at rationals and generalized harmonic numbers","translated_title":"","metadata":{"grobid_abstract":"Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer and rational values of arguments to certain rapidly converging series, where some generalized harmonic numbers appear. Most of the results of the paper can be derived from the recent, more advanced results, on the properties of Arakawa-Kaneko zeta functions. We derive our results directly, by solving simple recursions. The form of mentioned above generalized harmonic numbers carries information, about the values of the arguments of Hurwitz function. In particular we prove: 8k 2 N : (k; 1) = (k) = 2 k 1 2 k 1 1","publication_date":{"day":12,"month":1,"year":2015,"errors":{}},"publication_name":"Mathematica Eterna","grobid_abstract_attachment_id":115484121},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282237/On_connection_between_values_of_Riemann_zeta_function_at_rationals_and_generalized_harmonic_numbers","translated_internal_url":"","created_at":"2024-05-30T11:35:36.457-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484121,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484121/thumbnails/1.jpg","file_name":"MA5_4_5.pdf","download_url":"https://www.academia.edu/attachments/115484121/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_connection_between_values_of_Riemann.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484121/MA5_4_5-libre.pdf?1717097611=\u0026response-content-disposition=attachment%3B+filename%3DOn_connection_between_values_of_Riemann.pdf\u0026Expires=1734522074\u0026Signature=ad5iZP0vFj-t1pH3Sg3Qr7jkTqAhYp5wOjzfKv9KHKEKMChpIhaZQoICUayIx4lkCHbE3tAsjTgHgmH9l8HTCotRU3ULygRAAOldLlmTLdCjAutKg4G1uRvT~CvTkTmrkhWk1tMHBOsd956Ghbb-HuJJy5pQi3sQw0JApuZnfMLx8o2ctIkXTyJgc3XmPoddilEkfeV5WZ7z3Oi06IKH0fosPgLa4juLUC76Me6fP1EYJ4KJq5v1EQQW0Wc5E7I6gRGOm7BeqG6QBl6qnP-MVkVbdMR1NelMXDcHfQCKe2ZTOzjLGVx0KQufFx2kseefPE1arnfvsTwFyP0rCpVilw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_connection_between_values_of_Riemann_zeta_function_at_rationals_and_generalized_harmonic_numbers","translated_slug":"","page_count":10,"language":"en","content_type":"Work","summary":"Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer and rational values of arguments to certain rapidly converging series, where some generalized harmonic numbers appear. Most of the results of the paper can be derived from the recent, more advanced results, on the properties of Arakawa-Kaneko zeta functions. We derive our results directly, by solving simple recursions. The form of mentioned above generalized harmonic numbers carries information, about the values of the arguments of Hurwitz function. In particular we prove: 8k 2 N : (k; 1) = (k) = 2 k 1 2 k 1 1","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484121,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484121/thumbnails/1.jpg","file_name":"MA5_4_5.pdf","download_url":"https://www.academia.edu/attachments/115484121/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_connection_between_values_of_Riemann.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484121/MA5_4_5-libre.pdf?1717097611=\u0026response-content-disposition=attachment%3B+filename%3DOn_connection_between_values_of_Riemann.pdf\u0026Expires=1734522074\u0026Signature=ad5iZP0vFj-t1pH3Sg3Qr7jkTqAhYp5wOjzfKv9KHKEKMChpIhaZQoICUayIx4lkCHbE3tAsjTgHgmH9l8HTCotRU3ULygRAAOldLlmTLdCjAutKg4G1uRvT~CvTkTmrkhWk1tMHBOsd956Ghbb-HuJJy5pQi3sQw0JApuZnfMLx8o2ctIkXTyJgc3XmPoddilEkfeV5WZ7z3Oi06IKH0fosPgLa4juLUC76Me6fP1EYJ4KJq5v1EQQW0Wc5E7I6gRGOm7BeqG6QBl6qnP-MVkVbdMR1NelMXDcHfQCKe2ZTOzjLGVx0KQufFx2kseefPE1arnfvsTwFyP0rCpVilw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":229028,"name":"riemann Hypothesis","url":"https://www.academia.edu/Documents/in/riemann_Hypothesis"},{"id":1494623,"name":"Real Number","url":"https://www.academia.edu/Documents/in/Real_Number"}],"urls":[{"id":42488747,"url":"https://www.longdom.org/articles/on-connection-between-values-of-riemann-zeta-function-at-rationals-and-generalized-harmonic-numbers.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282236"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes"><img alt="Research paper thumbnail of q-Wiener and related processes. A continuous time generalization of Bryc processes" class="work-thumbnail" src="https://attachments.academia-assets.com/115484111/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes">q-Wiener and related processes. A continuous time generalization of Bryc processes</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jul 14, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We collect, scattered through literature, as well as we prove some new properties of two Markov p...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X = (Xt) t≥0 called q−Wiener) depends on one parameter q ∈ (−1, 1] and of the second one (say Y = (Yt) t∈R called (α, q)− Ornstein-Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles Wiener process in the sense that for q = 1 it is Wiener process but also that for |q| < 1 and ∀n ≥ 1 : t n/2 Hn Xt/ √ t|q , where (Hn) n≥0 are the so called q−Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein-Uhlenbeck process. For q = 1 it is a classical OU process. For |q| < 1 it is also stationary with correlation function equal to exp(−α|t − s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a71bb2cf2f7c2e07ecb0b5d2621ce6cb" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484111,"asset_id":120282236,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484111/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282236"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282236"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282236; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282236]").text(description); $(".js-view-count[data-work-id=120282236]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282236; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282236']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282236, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a71bb2cf2f7c2e07ecb0b5d2621ce6cb" } } $('.js-work-strip[data-work-id=120282236]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282236,"title":"q-Wiener and related processes. A continuous time generalization of Bryc processes","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X = (Xt) t≥0 called q−Wiener) depends on one parameter q ∈ (−1, 1] and of the second one (say Y = (Yt) t∈R called (α, q)− Ornstein-Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles Wiener process in the sense that for q = 1 it is Wiener process but also that for |q| \u003c 1 and ∀n ≥ 1 : t n/2 Hn Xt/ √ t|q , where (Hn) n≥0 are the so called q−Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein-Uhlenbeck process. For q = 1 it is a classical OU process. For |q| \u003c 1 it is also stationary with correlation function equal to exp(−α|t − s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.","publication_date":{"day":14,"month":7,"year":2005,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":115484111},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes","translated_internal_url":"","created_at":"2024-05-30T11:35:36.268-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484111,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484111/thumbnails/1.jpg","file_name":"0507303.pdf","download_url":"https://www.academia.edu/attachments/115484111/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"q_Wiener_and_related_processes_A_continu.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484111/0507303-libre.pdf?1717097627=\u0026response-content-disposition=attachment%3B+filename%3Dq_Wiener_and_related_processes_A_continu.pdf\u0026Expires=1734522074\u0026Signature=cFJgekt1Bz0gVAbes-PLPkFHsrXrNivfKBetwP86Dmp4nSrHl5ImHOBTDv5PSD3ctL-oXs593FEiQ~5c9mCjAITw0XQBJwsgBO7bWyigFAEiu9YYTNKWyop60F5ryyELIMMVfXrMTHd-7G-3UPca-P0y6LAhvhU4kifq-9i1ixnnUeXbBViogGWHxmdBuyTcz~PXiK-hfC7neNpCX8jdoLp41PmVT6d5ivW11r8efZgoZF6CW2mKg8GgWKPuWGzxvBKCk1ceSyj4rY8m7oit~3budDEdC6JHqsTauLa4GyhpIyoJ2TL6QQJXUCMqnh1e7SNTbTmGORlnPnZ966IYwg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes","translated_slug":"","page_count":25,"language":"en","content_type":"Work","summary":"We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X = (Xt) t≥0 called q−Wiener) depends on one parameter q ∈ (−1, 1] and of the second one (say Y = (Yt) t∈R called (α, q)− Ornstein-Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles Wiener process in the sense that for q = 1 it is Wiener process but also that for |q| \u003c 1 and ∀n ≥ 1 : t n/2 Hn Xt/ √ t|q , where (Hn) n≥0 are the so called q−Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein-Uhlenbeck process. For q = 1 it is a classical OU process. For |q| \u003c 1 it is also stationary with correlation function equal to exp(−α|t − s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484111,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484111/thumbnails/1.jpg","file_name":"0507303.pdf","download_url":"https://www.academia.edu/attachments/115484111/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"q_Wiener_and_related_processes_A_continu.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484111/0507303-libre.pdf?1717097627=\u0026response-content-disposition=attachment%3B+filename%3Dq_Wiener_and_related_processes_A_continu.pdf\u0026Expires=1734522074\u0026Signature=cFJgekt1Bz0gVAbes-PLPkFHsrXrNivfKBetwP86Dmp4nSrHl5ImHOBTDv5PSD3ctL-oXs593FEiQ~5c9mCjAITw0XQBJwsgBO7bWyigFAEiu9YYTNKWyop60F5ryyELIMMVfXrMTHd-7G-3UPca-P0y6LAhvhU4kifq-9i1ixnnUeXbBViogGWHxmdBuyTcz~PXiK-hfC7neNpCX8jdoLp41PmVT6d5ivW11r8efZgoZF6CW2mKg8GgWKPuWGzxvBKCk1ceSyj4rY8m7oit~3budDEdC6JHqsTauLa4GyhpIyoJ2TL6QQJXUCMqnh1e7SNTbTmGORlnPnZ966IYwg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":115484110,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484110/thumbnails/1.jpg","file_name":"0507303.pdf","download_url":"https://www.academia.edu/attachments/115484110/download_file","bulk_download_file_name":"q_Wiener_and_related_processes_A_continu.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484110/0507303-libre.pdf?1717097623=\u0026response-content-disposition=attachment%3B+filename%3Dq_Wiener_and_related_processes_A_continu.pdf\u0026Expires=1734522074\u0026Signature=ar-MChd~HMnik2pkRIkrW9jYJvbxkok-EZdaBzRTtx2MjvZMKtDLJx48mt04xxQMjYQXdwoGXKLAQg6QOpOJ8e-Ax9B3QLbRKiZb84U8GB708hCRLc4AaovmIeR2laQUIbC~BSY3cn~7OcPElivlIsGkATbXiblFtvJOc~hizFZZDQhl0py8aJC0sGOBfmsoTQdqsdVr9sLM-WhX9JWRD-nzBOpiBt3t08Fcl201kmWQYFfDflx3Cvgk-bL9uDB4sn1OMn6ujZD~88eh4w6u1fXqqSMCBP43REXqBKS64aztz7bs7jNJ~dDe1Y5tCG14JH4AR9Z-EOZVzLk4NtG-Wg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":389829,"name":"Gaussian distribution","url":"https://www.academia.edu/Documents/in/Gaussian_distribution"},{"id":679783,"name":"Boolean Satisfiability","url":"https://www.academia.edu/Documents/in/Boolean_Satisfiability"},{"id":741671,"name":"Markov Process","url":"https://www.academia.edu/Documents/in/Markov_Process"},{"id":2382100,"name":"Correlation function","url":"https://www.academia.edu/Documents/in/Correlation_function"},{"id":3115633,"name":"Discrete time","url":"https://www.academia.edu/Documents/in/Discrete_time"}],"urls":[{"id":42488746,"url":"http://arxiv.org/pdf/math/0507303"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282234"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282234/On_probabilistic_aspects_of_Chebyshev_polynomials"><img alt="Research paper thumbnail of On probabilistic aspects of Chebyshev polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/115484108/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282234/On_probabilistic_aspects_of_Chebyshev_polynomials">On probabilistic aspects of Chebyshev polynomials</a></div><div class="wp-workCard_item"><span>Statistics & Probability Letters</span><span>, Feb 1, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We examine the properties of distributions with the density of the form: 2Anc n−2 √ c 2 −x 2 π n ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We examine the properties of distributions with the density of the form: 2Anc n−2 √ c 2 −x 2 π n j=1 (c(1+a 2 j)−2a j x) , where c, a 1 ,. .. , an are some parameters and An a suitable constant. We find general forms of An, of k−th moment and of k−th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey-Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a 1 ,. .. , an forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6. n j=1 (c(1 + a 2 j) − 2a j x) , defined for n ≥ 0, |x| ≤ c, with c > 0, |a j | < 1, j = 1,. .. , n. Here A n is a normalizing constant being the function of parameters a 1 ,. .. , a n. We will call this family generalized Kesten-McKay distributions. The name is justified by the fact that the distribution with the following density: (1.2) f KMK2 (x|2/a, a, −a) = v 4(v − 1) − x 2 2π(v 2 − x 2) , where v = 1+1/a 2 and |a| < 1 has been defined, described and what is more, derived in [5] or [9]. Then the name Kesten-McKay distribution has been attributed to this distribution in the literature that appeared after 1981. Thus, it is justified to call the distribution defined by (1.1) a generalized Kesten-McKay (GKM) distribution. Note also that for n = 0 the distribution with the density f KMK0 (x|c) becomes Wigner or semicircle distribution with parameter c.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3decd0381c45891822fdf4908c89aed0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484108,"asset_id":120282234,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484108/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282234"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282234"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282234; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282234]").text(description); $(".js-view-count[data-work-id=120282234]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282234; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282234']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282234, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "3decd0381c45891822fdf4908c89aed0" } } $('.js-work-strip[data-work-id=120282234]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282234,"title":"On probabilistic aspects of Chebyshev polynomials","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We examine the properties of distributions with the density of the form: 2Anc n−2 √ c 2 −x 2 π n j=1 (c(1+a 2 j)−2a j x) , where c, a 1 ,. .. , an are some parameters and An a suitable constant. We find general forms of An, of k−th moment and of k−th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey-Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a 1 ,. .. , an forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6. n j=1 (c(1 + a 2 j) − 2a j x) , defined for n ≥ 0, |x| ≤ c, with c \u003e 0, |a j | \u003c 1, j = 1,. .. , n. Here A n is a normalizing constant being the function of parameters a 1 ,. .. , a n. We will call this family generalized Kesten-McKay distributions. The name is justified by the fact that the distribution with the following density: (1.2) f KMK2 (x|2/a, a, −a) = v 4(v − 1) − x 2 2π(v 2 − x 2) , where v = 1+1/a 2 and |a| \u003c 1 has been defined, described and what is more, derived in [5] or [9]. Then the name Kesten-McKay distribution has been attributed to this distribution in the literature that appeared after 1981. Thus, it is justified to call the distribution defined by (1.1) a generalized Kesten-McKay (GKM) distribution. Note also that for n = 0 the distribution with the density f KMK0 (x|c) becomes Wigner or semicircle distribution with parameter c.","publication_date":{"day":1,"month":2,"year":2019,"errors":{}},"publication_name":"Statistics \u0026 Probability Letters","grobid_abstract_attachment_id":115484108},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282234/On_probabilistic_aspects_of_Chebyshev_polynomials","translated_internal_url":"","created_at":"2024-05-30T11:35:34.944-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484108,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484108/thumbnails/1.jpg","file_name":"1507.pdf","download_url":"https://www.academia.edu/attachments/115484108/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_probabilistic_aspects_of_Chebyshev_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484108/1507-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DOn_probabilistic_aspects_of_Chebyshev_po.pdf\u0026Expires=1734522074\u0026Signature=fkU8SF9oX4lUXQaSeCcs0gJEomCmpXPl8PpeKN0eroslHWdDagjkAeEXv7f891l8P73fz324TaluqgY-uxclxDUrtfndEj1OV5M6PQ3Xvpzc951rb2XV-qP6~78wBXOGn5p-iqTvtfEHfO8WO-oNhJonV6Xp~FxNn9hqMvqIzKMuO95kyTDrV-qhiQ-OjX2nCEaheZQf4HSdzyjHWAneQTf95a64hM2awchjNXKyqemjTJ0DbByXM25aCNRxsovz9Ge-1-XZL5Lv1IbIUQ~Xmu9DVJR-ISWDpk0JAkOkgRHf7kVICXi4k6E-QOdMkUpST96ZjQZrSQYo0Ulscqnexw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_probabilistic_aspects_of_Chebyshev_polynomials","translated_slug":"","page_count":13,"language":"en","content_type":"Work","summary":"We examine the properties of distributions with the density of the form: 2Anc n−2 √ c 2 −x 2 π n j=1 (c(1+a 2 j)−2a j x) , where c, a 1 ,. .. , an are some parameters and An a suitable constant. We find general forms of An, of k−th moment and of k−th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey-Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a 1 ,. .. , an forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6. n j=1 (c(1 + a 2 j) − 2a j x) , defined for n ≥ 0, |x| ≤ c, with c \u003e 0, |a j | \u003c 1, j = 1,. .. , n. Here A n is a normalizing constant being the function of parameters a 1 ,. .. , a n. We will call this family generalized Kesten-McKay distributions. The name is justified by the fact that the distribution with the following density: (1.2) f KMK2 (x|2/a, a, −a) = v 4(v − 1) − x 2 2π(v 2 − x 2) , where v = 1+1/a 2 and |a| \u003c 1 has been defined, described and what is more, derived in [5] or [9]. Then the name Kesten-McKay distribution has been attributed to this distribution in the literature that appeared after 1981. Thus, it is justified to call the distribution defined by (1.1) a generalized Kesten-McKay (GKM) distribution. Note also that for n = 0 the distribution with the density f KMK0 (x|c) becomes Wigner or semicircle distribution with parameter c.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484108,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484108/thumbnails/1.jpg","file_name":"1507.pdf","download_url":"https://www.academia.edu/attachments/115484108/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_probabilistic_aspects_of_Chebyshev_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484108/1507-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DOn_probabilistic_aspects_of_Chebyshev_po.pdf\u0026Expires=1734522074\u0026Signature=fkU8SF9oX4lUXQaSeCcs0gJEomCmpXPl8PpeKN0eroslHWdDagjkAeEXv7f891l8P73fz324TaluqgY-uxclxDUrtfndEj1OV5M6PQ3Xvpzc951rb2XV-qP6~78wBXOGn5p-iqTvtfEHfO8WO-oNhJonV6Xp~FxNn9hqMvqIzKMuO95kyTDrV-qhiQ-OjX2nCEaheZQf4HSdzyjHWAneQTf95a64hM2awchjNXKyqemjTJ0DbByXM25aCNRxsovz9Ge-1-XZL5Lv1IbIUQ~Xmu9DVJR-ISWDpk0JAkOkgRHf7kVICXi4k6E-QOdMkUpST96ZjQZrSQYo0Ulscqnexw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":115484109,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484109/thumbnails/1.jpg","file_name":"1507.pdf","download_url":"https://www.academia.edu/attachments/115484109/download_file","bulk_download_file_name":"On_probabilistic_aspects_of_Chebyshev_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484109/1507-libre.pdf?1717097619=\u0026response-content-disposition=attachment%3B+filename%3DOn_probabilistic_aspects_of_Chebyshev_po.pdf\u0026Expires=1734522074\u0026Signature=EsiFNiputI8ee4QzHgP9btvE~bWfIvXgHk0BU53dEJP2x1KDUytqXYhQ7-sJ~fjFOZNKus2oFnr-I-Kn12kMt-sDuO7gEjeeqNx5PfiXZnrXynpEW-DGIhelfslrsMy7IfVSExoFxf4o1G0OZ7NRq51r5YAuUYg5KHBXrSfq8kE4rwRknnTlvqjqION-IJkHyjFh8hVBpzKxMeKGgAfaojxHnMgvQFeYxhTfpdOucAgB0HSZq7MKDQA93BKm5tW3Jbsd4joHnNLkLmqtNe18QDqUsBLRHkJtvrG3d6nQ6~vUcRQ2KBN4XYNh3KcRVUiyZqnv6Z5DJwap8kyf0i~uMQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":747,"name":"Econometrics","url":"https://www.academia.edu/Documents/in/Econometrics"},{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"},{"id":12026,"name":"Orthogonal polynomials","url":"https://www.academia.edu/Documents/in/Orthogonal_polynomials"},{"id":2534646,"name":"Chebyshev polynomials","url":"https://www.academia.edu/Documents/in/Chebyshev_polynomials"},{"id":2730277,"name":"Chebyshev filter","url":"https://www.academia.edu/Documents/in/Chebyshev_filter"}],"urls":[{"id":42488745,"url":"http://arxiv.org/pdf/1507.03191"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282233"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282233/On_three_dimensional_multivariate_version_of_q_Normal_distribution_and_probabilistic_interpretations_of_Askey_Wilson_Al_Salam_Chihara_and_q_ultraspherical_polynomials"><img alt="Research paper thumbnail of On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey–Wilson, Al-Salam–Chihara and q-ultraspherical polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/115484122/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282233/On_three_dimensional_multivariate_version_of_q_Normal_distribution_and_probabilistic_interpretations_of_Askey_Wilson_Al_Salam_Chihara_and_q_ultraspherical_polynomials">On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey–Wilson, Al-Salam–Chihara and q-ultraspherical polynomials</a></div><div class="wp-workCard_item"><span>Journal of Mathematical Analysis and Applications</span><span>, Jun 1, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study properties of compactly supported, 4 parameter (ρ 12 , ρ 23 , ρ 13 , q) ∈ (−1, 1) ×4 fam...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We study properties of compactly supported, 4 parameter (ρ 12 , ρ 23 , ρ 13 , q) ∈ (−1, 1) ×4 family of continuous type 3 dimensional distributions, that have the property that for q → 1 − this family tends to some 3 dimensional Normal distribution. For q = 0 we deal with 3 dimensional generalization of Kesten-McKay distribution. In a very special case when ρ 12 ρ 13 ρ 23 = q all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order n and are polynomials of the same order n in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey-Wilson, Al-Salam-Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="05f38fdda0ae638849d9544c63295f44" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484122,"asset_id":120282233,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484122/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282233"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282233"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282233; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282233]").text(description); $(".js-view-count[data-work-id=120282233]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282233; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282233']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282233, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "05f38fdda0ae638849d9544c63295f44" } } $('.js-work-strip[data-work-id=120282233]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282233,"title":"On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey–Wilson, Al-Salam–Chihara and q-ultraspherical polynomials","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We study properties of compactly supported, 4 parameter (ρ 12 , ρ 23 , ρ 13 , q) ∈ (−1, 1) ×4 family of continuous type 3 dimensional distributions, that have the property that for q → 1 − this family tends to some 3 dimensional Normal distribution. For q = 0 we deal with 3 dimensional generalization of Kesten-McKay distribution. In a very special case when ρ 12 ρ 13 ρ 23 = q all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order n and are polynomials of the same order n in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey-Wilson, Al-Salam-Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials.","publication_date":{"day":1,"month":6,"year":2019,"errors":{}},"publication_name":"Journal of Mathematical Analysis and Applications","grobid_abstract_attachment_id":115484122},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282233/On_three_dimensional_multivariate_version_of_q_Normal_distribution_and_probabilistic_interpretations_of_Askey_Wilson_Al_Salam_Chihara_and_q_ultraspherical_polynomials","translated_internal_url":"","created_at":"2024-05-30T11:35:34.744-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484122,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484122/thumbnails/1.jpg","file_name":"1712.pdf","download_url":"https://www.academia.edu/attachments/115484122/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_three_dimensional_multivariate_versio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484122/1712-libre.pdf?1717097617=\u0026response-content-disposition=attachment%3B+filename%3DOn_three_dimensional_multivariate_versio.pdf\u0026Expires=1734522074\u0026Signature=dIetTf7MBAUbpbPr9IJHIb~LfmMucc~lQd2vwRTUBzAzfHZgyYtJ83hsp-0zg7-0vwp2HspxocdI~cONIvciaZpmjDCkhcDt075EYsJLGQgE0d9Asm6dOgmO166k~mrTF8YviclqFXzRshS84BYV9BZTiEBYEmhweQBBcoLzSbPh~v05lGOTVRIkq5i8AuCQ4G3uUZ2u-SCvE8ctJj~I3x8K7Q3UCSJ~mCyVM06ij6BdpiIV-lLD8tEKU4ghXwffIgY9aEawWowRs8FUku~oknNARxtGwkCJgDBZc7bkTtQyUUZt2wi-sI31hV6ZIyapR-cLM-PwEQ6Z1y-~y1PTqA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_three_dimensional_multivariate_version_of_q_Normal_distribution_and_probabilistic_interpretations_of_Askey_Wilson_Al_Salam_Chihara_and_q_ultraspherical_polynomials","translated_slug":"","page_count":15,"language":"en","content_type":"Work","summary":"We study properties of compactly supported, 4 parameter (ρ 12 , ρ 23 , ρ 13 , q) ∈ (−1, 1) ×4 family of continuous type 3 dimensional distributions, that have the property that for q → 1 − this family tends to some 3 dimensional Normal distribution. For q = 0 we deal with 3 dimensional generalization of Kesten-McKay distribution. In a very special case when ρ 12 ρ 13 ρ 23 = q all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order n and are polynomials of the same order n in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey-Wilson, Al-Salam-Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484122,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484122/thumbnails/1.jpg","file_name":"1712.pdf","download_url":"https://www.academia.edu/attachments/115484122/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_three_dimensional_multivariate_versio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484122/1712-libre.pdf?1717097617=\u0026response-content-disposition=attachment%3B+filename%3DOn_three_dimensional_multivariate_versio.pdf\u0026Expires=1734522074\u0026Signature=dIetTf7MBAUbpbPr9IJHIb~LfmMucc~lQd2vwRTUBzAzfHZgyYtJ83hsp-0zg7-0vwp2HspxocdI~cONIvciaZpmjDCkhcDt075EYsJLGQgE0d9Asm6dOgmO166k~mrTF8YviclqFXzRshS84BYV9BZTiEBYEmhweQBBcoLzSbPh~v05lGOTVRIkq5i8AuCQ4G3uUZ2u-SCvE8ctJj~I3x8K7Q3UCSJ~mCyVM06ij6BdpiIV-lLD8tEKU4ghXwffIgY9aEawWowRs8FUku~oknNARxtGwkCJgDBZc7bkTtQyUUZt2wi-sI31hV6ZIyapR-cLM-PwEQ6Z1y-~y1PTqA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":12026,"name":"Orthogonal polynomials","url":"https://www.academia.edu/Documents/in/Orthogonal_polynomials"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":346242,"name":"Hermite Polynomials","url":"https://www.academia.edu/Documents/in/Hermite_Polynomials"},{"id":518958,"name":"Mathematical Analysis and Applications","url":"https://www.academia.edu/Documents/in/Mathematical_Analysis_and_Applications"},{"id":1237788,"name":"Electrical And Electronic Engineering","url":"https://www.academia.edu/Documents/in/Electrical_And_Electronic_Engineering"}],"urls":[{"id":42488744,"url":"https://doi.org/10.1016/j.jmaa.2019.02.002"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282232"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282232/Elliptically_contoured_random_variables_and_their_application_to_the_extension_of_the_Kalman_filter"><img alt="Research paper thumbnail of Elliptically contoured random variables and their application to the extension of the Kalman filter" class="work-thumbnail" src="https://attachments.academia-assets.com/115484124/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282232/Elliptically_contoured_random_variables_and_their_application_to_the_extension_of_the_Kalman_filter">Elliptically contoured random variables and their application to the extension of the Kalman filter</a></div><div class="wp-workCard_item"><span>Computers & mathematics with applications</span><span>, 1990</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We summarize some properties of elhptlcall} contoured (e c.) random xanables and indicate their p...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We summarize some properties of elhptlcall} contoured (e c.) random xanables and indicate their possible apphcation in engineering practice. In parttcular after presenting some nex~ results ~e appl} them to modify the Kalman filter. The modified ',erslon is robust against the extenston of the no~ses' model from normal to ellipttcall~ contoured ,%'ole on notation All vector are columns x T means transposition of xector .v lhence v T is a row vector). Matrices and vectors are multiplied according to usual rules. If</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="44a8157f511a27f0c656f8a558559ec2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484124,"asset_id":120282232,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484124/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282232"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282232"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282232; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282232]").text(description); $(".js-view-count[data-work-id=120282232]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282232; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282232']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282232, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "44a8157f511a27f0c656f8a558559ec2" } } $('.js-work-strip[data-work-id=120282232]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282232,"title":"Elliptically contoured random variables and their application to the extension of the Kalman filter","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We summarize some properties of elhptlcall} contoured (e c.) random xanables and indicate their possible apphcation in engineering practice. 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If","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484124,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484124/thumbnails/1.jpg","file_name":"81946991.pdf","download_url":"https://www.academia.edu/attachments/115484124/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Elliptically_contoured_random_variables.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484124/81946991-libre.pdf?1717097613=\u0026response-content-disposition=attachment%3B+filename%3DElliptically_contoured_random_variables.pdf\u0026Expires=1734522074\u0026Signature=ZIUWWmKrfo~iK5IVfvjDnMu-uRZFuP39gQ50WKc8Hx5uTojXnbKVqiT0r5hSm23Jqv3AGCoQszfmxTRsfbSCMkToRZd8AOKPdAabgFbDDu1f-cWQyVwu~ZDAbEnZncbnzKy7uIa4wSMEPeDtbP3HBx~t6mMdf6Ju0ylucekg7T1GLAHuGWj-iYbnGRO7xdmo1UhINpwVDwSQQjhHGyT5~R9zKX6gi61kcN0tj8Ndzsmdap1s35X7MHuJW3SdEaupFMef9bGlGOZCc0axanWNl2dfqI-gEK6gM2I9-C-le6mFyodWVBAENNgrAzK4KiX7~GtC4vB5tH84DXE8eKzp5g__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":49146,"name":"Kalman Filter","url":"https://www.academia.edu/Documents/in/Kalman_Filter"},{"id":80414,"name":"Mathematical Sciences","url":"https://www.academia.edu/Documents/in/Mathematical_Sciences"},{"id":171894,"name":"Computers and Mathematics with Applications 59 (2010) 35783582","url":"https://www.academia.edu/Documents/in/Computers_and_Mathematics_with_Applications_59_2010_35783582"},{"id":611819,"name":"Discrete random variable","url":"https://www.academia.edu/Documents/in/Discrete_random_variable"}],"urls":[{"id":42488743,"url":"https://doi.org/10.1016/0898-1221(90)90008-8"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282231"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282231/Gaussian_Distributions_Simplifications_and_Simulations"><img alt="Research paper thumbnail of Gaussian Distributions: Simplifications and Simulations" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282231/Gaussian_Distributions_Simplifications_and_Simulations">Gaussian Distributions: Simplifications and Simulations</a></div><div class="wp-workCard_item"><span>Journal of Probability and Statistics</span><span>, 2009</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We present some properties of measures (-Gaussian) that orthogonalize the set of -Hermite polynom...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We present some properties of measures (-Gaussian) that orthogonalize the set of -Hermite polynomials. We also present an algorithm for simulating i.i.d. sequences of random variables having -Gaussian distribution.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282231"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282231"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282231; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282231]").text(description); $(".js-view-count[data-work-id=120282231]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282231; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282231']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282231, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=120282231]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282231,"title":"Gaussian Distributions: Simplifications and Simulations","translated_title":"","metadata":{"abstract":"We present some properties of measures (-Gaussian) that orthogonalize the set of -Hermite polynomials. 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We also present an algorithm for simulating i.i.d. sequences of random variables having -Gaussian distribution.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":16460,"name":"Statistical Physics","url":"https://www.academia.edu/Documents/in/Statistical_Physics"},{"id":342314,"name":"Gaussian","url":"https://www.academia.edu/Documents/in/Gaussian"},{"id":1523776,"name":"Probability statistics","url":"https://www.academia.edu/Documents/in/Probability_statistics"}],"urls":[{"id":42488742,"url":"https://doi.org/10.1155/2009/752430"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282230"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282230/On_the_structure_and_probabilistic_interpretation_of_Askey_Wilson_densities_and_polynomials_with_complex_parameters"><img alt="Research paper thumbnail of On the structure and probabilistic interpretation of Askey–Wilson densities and polynomials with complex parameters" class="work-thumbnail" src="https://attachments.academia-assets.com/115484129/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282230/On_the_structure_and_probabilistic_interpretation_of_Askey_Wilson_densities_and_polynomials_with_complex_parameters">On the structure and probabilistic interpretation of Askey–Wilson densities and polynomials with complex parameters</a></div><div class="wp-workCard_item"><span>Journal of Functional Analysis</span><span>, Aug 1, 2011</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We give equivalent forms of the Askey-Wilson polynomials expressing them with the help of the Al-...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We give equivalent forms of the Askey-Wilson polynomials expressing them with the help of the Al-Salam-Chihara polynomials. After restricting parameters of the Askey-Wilson polynomials to complex conjugate pairs we expand the Askey-Wilson weight function in the series similar to the Poisson-Mehler expansion formula and give its probabilistic interpretation. In particular this result can be used to calculate explicit forms of 'q-Hermite' moments of the Askey-Wilson density, hence enabling calculation of all moments of the Askey-Wilson density. On the way (by setting certain parameter q to 0) we get some formulae useful in the rapidly developing so called 'free probability'.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="76a7b5cbaa2825cb9cdb166e9e01727c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484129,"asset_id":120282230,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484129/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282230"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282230"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282230; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282230]").text(description); $(".js-view-count[data-work-id=120282230]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282230; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282230']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282230, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "76a7b5cbaa2825cb9cdb166e9e01727c" } } $('.js-work-strip[data-work-id=120282230]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282230,"title":"On the structure and probabilistic interpretation of Askey–Wilson densities and polynomials with complex parameters","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We give equivalent forms of the Askey-Wilson polynomials expressing them with the help of the Al-Salam-Chihara polynomials. 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On the way (by setting certain parameter q to 0) we get some formulae useful in the rapidly developing so called 'free probability'.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484129,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484129/thumbnails/1.jpg","file_name":"1011.pdf","download_url":"https://www.academia.edu/attachments/115484129/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_structure_and_probabilistic_inter.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484129/1011-libre.pdf?1717097623=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_structure_and_probabilistic_inter.pdf\u0026Expires=1734522074\u0026Signature=GXQy9mEptNlDygBqTlXOV-VD9Uy2IWN~BagKXC3TMFDc4q76tB6KjFSQ9g3zRBiJ-Y7VIhh9h77VdrRhfR0UxaJyObowTCkE7G~4zl-FTHmGfDaT-QxdIzOwGWskZ0FuVZVO1EY60CjblYBCx8248h8rOuUhxNqx9kQTpHaTYxFsoj2UVqZkkdBGkQyc4EtsZhPHfKK0~j1af~sm2Br9xkjgCKPianZqHghSeOmWMIMSCYxA1MmD4otmpOatTRY852B6AEQi0sATXXQF~1gDJmoH-3H95LbEwGPEmsGMIrmu3Vga43qzLunZShqzzaBtxa3GuNr2QfizdzZA57mbjw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":371,"name":"Functional Analysis","url":"https://www.academia.edu/Documents/in/Functional_Analysis"},{"id":12026,"name":"Orthogonal polynomials","url":"https://www.academia.edu/Documents/in/Orthogonal_polynomials"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":346242,"name":"Hermite Polynomials","url":"https://www.academia.edu/Documents/in/Hermite_Polynomials"},{"id":434088,"name":"Free Probability","url":"https://www.academia.edu/Documents/in/Free_Probability"},{"id":2534646,"name":"Chebyshev polynomials","url":"https://www.academia.edu/Documents/in/Chebyshev_polynomials"}],"urls":[{"id":42488741,"url":"https://doi.org/10.1016/j.jfa.2011.04.002"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282229"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282229/Askey_Wilson_Integral_and_its_Generalizations"><img alt="Research paper thumbnail of Askey--Wilson Integral and its Generalizations" class="work-thumbnail" src="https://attachments.academia-assets.com/115484107/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282229/Askey_Wilson_Integral_and_its_Generalizations">Askey--Wilson Integral and its Generalizations</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 20, 2011</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We expand the Askey-Wilson (AW) density in a series of products of continuous q−Hermite polynomia...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We expand the Askey-Wilson (AW) density in a series of products of continuous q−Hermite polynomials times the density that makes these polynomials orthogonal. As a by-product we obtain the value of the AW integral as well as the values of integrals of q−Hermite polynomial times the AW density (q−Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. As a result we obtain system of recurrences that if solved successfully gives a sequence of generalized AW densities with more and more parameters.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="276c20e8b29a856362d73ab51ec8cd1c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484107,"asset_id":120282229,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484107/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282229"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282229"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282229; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282229]").text(description); $(".js-view-count[data-work-id=120282229]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282229; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282229']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282229, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "276c20e8b29a856362d73ab51ec8cd1c" } } $('.js-work-strip[data-work-id=120282229]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282229,"title":"Askey--Wilson Integral and its Generalizations","translated_title":"","metadata":{"publisher":"Cornell University","ai_title_tag":"Generalizations of Askey-Wilson Integral and Densities","grobid_abstract":"We expand the Askey-Wilson (AW) density in a series of products of continuous q−Hermite polynomials times the density that makes these polynomials orthogonal. 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As a by-product we obtain the value of the AW integral as well as the values of integrals of q−Hermite polynomial times the AW density (q−Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. 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data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/123611904/Can_the_first_two_conditional_moments_identify_a_mean_square_differentiable_process"><img alt="Research paper thumbnail of Can the first two conditional moments identify a mean square differentiable process?" class="work-thumbnail" src="https://attachments.academia-assets.com/118003768/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/123611904/Can_the_first_two_conditional_moments_identify_a_mean_square_differentiable_process">Can the first two conditional moments identify a mean square differentiable process?</a></div><div class="wp-workCard_item"><span>Computers & mathematics with applications</span><span>, 1989</span></div><div class="wp-workCard_item wp-workCard--actions"><span 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href="https://www.academia.edu/120282242/Understanding_mathematics_of_Grover_s_algorithm"><img alt="Research paper thumbnail of Understanding mathematics of Grover’s algorithm" class="work-thumbnail" src="https://attachments.academia-assets.com/115484115/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282242/Understanding_mathematics_of_Grover_s_algorithm">Understanding mathematics of Grover’s algorithm</a></div><div class="wp-workCard_item"><span>Quantum Information Processing</span><span>, May 1, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We analyze the mathematical structure of the classical Grover's algorithm and put it within the f...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We analyze the mathematical structure of the classical Grover's algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one 'chosen' element (sometimes called a 'solution') of the dataset, but a set of m such 'chosen' elements (out of n > m). Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that 'marks,' by a suitable phase change ϕ, all these 'chosen' elements. In the first part of the paper, we construct a unique unitary operator that selects all 'chosen' elements in one step. The constructed operator is uniquely defined by the numbers ϕ and α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on α. In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, 'convenient' phase change ϕ, and by sequentially applying the so-constructed operator, we find the number of steps to find these 'chosen' elements with great probability. We apply this knowledge to study the generalizations of Grover's algorithm (m = 1, φ = π), which are of the form, the found previously, unitary operators.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a8aa8b6b117b72171216dcc6cd137617" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484115,"asset_id":120282242,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484115/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282242"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282242"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282242; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282242]").text(description); $(".js-view-count[data-work-id=120282242]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282242; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282242']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282242, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a8aa8b6b117b72171216dcc6cd137617" } } $('.js-work-strip[data-work-id=120282242]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282242,"title":"Understanding mathematics of Grover’s algorithm","translated_title":"","metadata":{"publisher":"Springer Science+Business Media","grobid_abstract":"We analyze the mathematical structure of the classical Grover's algorithm and put it within the framework of linear algebra over the complex numbers. 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In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, 'convenient' phase change ϕ, and by sequentially applying the so-constructed operator, we find the number of steps to find these 'chosen' elements with great probability. We apply this knowledge to study the generalizations of Grover's algorithm (m = 1, φ = π), which are of the form, the found previously, unitary operators.","publication_date":{"day":1,"month":5,"year":2021,"errors":{}},"publication_name":"Quantum Information Processing","grobid_abstract_attachment_id":115484115},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282242/Understanding_mathematics_of_Grover_s_algorithm","translated_internal_url":"","created_at":"2024-05-30T11:35:37.498-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484115,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484115/thumbnails/1.jpg","file_name":"s11128-021-03125-w.pdf","download_url":"https://www.academia.edu/attachments/115484115/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Understanding_mathematics_of_Grover_s_al.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484115/s11128-021-03125-w-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DUnderstanding_mathematics_of_Grover_s_al.pdf\u0026Expires=1734522074\u0026Signature=RnL9ogXdZGdw8f4ak1G5WlYzcZoYFq36FwZ-HeW~8d9CLPIdqIMF51NBjIFAdZdvj7GMz2Bqbzr76SzHTP4uLdeTiNrsC3gNdqKwVZHk4H4IC~PfrycqIWfWII3QM0PwZeNSfC8KXrTGNIJj4vYqaHC-mhWmLP9M2FHXMf6CA~K-VxloNJTOS-TSYJaXuPIsfqElPoWGPQgGVuT6Bt0ZEqTU1xVk715BSOAHF96yM2YgEFKRnVAoQ06UBHewnhoJRworQNEv6vahFzW8fACXUoZVnlu-~lfC3KJ4b5gOEVZ83YKs76YA3lDddM5wEuw~avX3AYCUxxGgKeHMLG38Yw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Understanding_mathematics_of_Grover_s_algorithm","translated_slug":"","page_count":21,"language":"en","content_type":"Work","summary":"We analyze the mathematical structure of the classical Grover's algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one 'chosen' element (sometimes called a 'solution') of the dataset, but a set of m such 'chosen' elements (out of n \u003e m). Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that 'marks,' by a suitable phase change ϕ, all these 'chosen' elements. In the first part of the paper, we construct a unique unitary operator that selects all 'chosen' elements in one step. The constructed operator is uniquely defined by the numbers ϕ and α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on α. In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, 'convenient' phase change ϕ, and by sequentially applying the so-constructed operator, we find the number of steps to find these 'chosen' elements with great probability. We apply this knowledge to study the generalizations of Grover's algorithm (m = 1, φ = π), which are of the form, the found previously, unitary operators.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484115,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484115/thumbnails/1.jpg","file_name":"s11128-021-03125-w.pdf","download_url":"https://www.academia.edu/attachments/115484115/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Understanding_mathematics_of_Grover_s_al.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484115/s11128-021-03125-w-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DUnderstanding_mathematics_of_Grover_s_al.pdf\u0026Expires=1734522074\u0026Signature=RnL9ogXdZGdw8f4ak1G5WlYzcZoYFq36FwZ-HeW~8d9CLPIdqIMF51NBjIFAdZdvj7GMz2Bqbzr76SzHTP4uLdeTiNrsC3gNdqKwVZHk4H4IC~PfrycqIWfWII3QM0PwZeNSfC8KXrTGNIJj4vYqaHC-mhWmLP9M2FHXMf6CA~K-VxloNJTOS-TSYJaXuPIsfqElPoWGPQgGVuT6Bt0ZEqTU1xVk715BSOAHF96yM2YgEFKRnVAoQ06UBHewnhoJRworQNEv6vahFzW8fACXUoZVnlu-~lfC3KJ4b5gOEVZ83YKs76YA3lDddM5wEuw~avX3AYCUxxGgKeHMLG38Yw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":318,"name":"Mathematical Physics","url":"https://www.academia.edu/Documents/in/Mathematical_Physics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":444,"name":"Quantum Computing","url":"https://www.academia.edu/Documents/in/Quantum_Computing"},{"id":518,"name":"Quantum Physics","url":"https://www.academia.edu/Documents/in/Quantum_Physics"},{"id":2640,"name":"Quantum Information","url":"https://www.academia.edu/Documents/in/Quantum_Information"},{"id":4199,"name":"Quantum Information Processing","url":"https://www.academia.edu/Documents/in/Quantum_Information_Processing"},{"id":26817,"name":"Algorithm","url":"https://www.academia.edu/Documents/in/Algorithm"}],"urls":[{"id":42488752,"url":"https://link.springer.com/content/pdf/10.1007/s11128-021-03125-w.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282241"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282241/Towards_a_q_analogue_of_the_Kibble_Slepian_formula_in_3_dimensions"><img alt="Research paper thumbnail of Towards a q-analogue of the Kibble–Slepian formula in 3 dimensions" class="work-thumbnail" src="https://attachments.academia-assets.com/115484123/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282241/Towards_a_q_analogue_of_the_Kibble_Slepian_formula_in_3_dimensions">Towards a q-analogue of the Kibble–Slepian formula in 3 dimensions</a></div><div class="wp-workCard_item"><span>Journal of Functional Analysis</span><span>, 2012</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The gener...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q−Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3 dimensional Normal distribution. We show that this is not the case. We indicate some values of the parameters and some compact set in R 3 of positive measure, such that the values of the extension of KS formula are on this set negative. Nevertheless we indicate other applications of so generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam-Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey-Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we indicate some new interesting properties of the Askey-Wilson polynomials with complex parameters. We also pose several open questions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f2786ecaed38008a146129a4f891087a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484123,"asset_id":120282241,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484123/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282241"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282241"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282241; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282241]").text(description); $(".js-view-count[data-work-id=120282241]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282241; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282241']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282241, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "f2786ecaed38008a146129a4f891087a" } } $('.js-work-strip[data-work-id=120282241]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282241,"title":"Towards a q-analogue of the Kibble–Slepian formula in 3 dimensions","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q−Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3 dimensional Normal distribution. We show that this is not the case. We indicate some values of the parameters and some compact set in R 3 of positive measure, such that the values of the extension of KS formula are on this set negative. Nevertheless we indicate other applications of so generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam-Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey-Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we indicate some new interesting properties of the Askey-Wilson polynomials with complex parameters. We also pose several open questions.","publication_date":{"day":null,"month":null,"year":2012,"errors":{}},"publication_name":"Journal of Functional Analysis","grobid_abstract_attachment_id":115484123},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282241/Towards_a_q_analogue_of_the_Kibble_Slepian_formula_in_3_dimensions","translated_internal_url":"","created_at":"2024-05-30T11:35:37.285-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484123,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484123/thumbnails/1.jpg","file_name":"1011.4929v5.pdf","download_url":"https://www.academia.edu/attachments/115484123/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Towards_a_q_analogue_of_the_Kibble_Slepi.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484123/1011.4929v5-libre.pdf?1717097620=\u0026response-content-disposition=attachment%3B+filename%3DTowards_a_q_analogue_of_the_Kibble_Slepi.pdf\u0026Expires=1734522074\u0026Signature=V7YwkUx8zn-QZ~3lgJtNXV6VPnjUtFPl2MTFKHtvrySYfoJ2oAZ7j5i-O1cSP31Ah6towz6oOktskmM72MvP7M85IvUAEqUdKYFQ3BAlf6mFcvE~lR4eO2yse4IFni79WgYgTyuNwXIXOC8Iz9OvoDuEmD~CCzlWBNJxNdeno3W74fVqxP0xPPZKVer0yJkYdt1sZdscWDKGS9eEZwqbzvGuEQ9dMlfi2e-~AoB4uH6UI-gxTYVzGvkxQIZC068sx0SmWrZRewhdHQAv4YjZFzm-M1MyKNrm3ZoivdFF8GJGMfY9~jsfqpPZd5UUejnj-dKd~-9sse4~wxraA98kTg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Towards_a_q_analogue_of_the_Kibble_Slepian_formula_in_3_dimensions","translated_slug":"","page_count":20,"language":"en","content_type":"Work","summary":"We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q−Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3 dimensional Normal distribution. We show that this is not the case. We indicate some values of the parameters and some compact set in R 3 of positive measure, such that the values of the extension of KS formula are on this set negative. Nevertheless we indicate other applications of so generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam-Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey-Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we indicate some new interesting properties of the Askey-Wilson polynomials with complex parameters. We also pose several open questions.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484123,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484123/thumbnails/1.jpg","file_name":"1011.4929v5.pdf","download_url":"https://www.academia.edu/attachments/115484123/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Towards_a_q_analogue_of_the_Kibble_Slepi.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484123/1011.4929v5-libre.pdf?1717097620=\u0026response-content-disposition=attachment%3B+filename%3DTowards_a_q_analogue_of_the_Kibble_Slepi.pdf\u0026Expires=1734522074\u0026Signature=V7YwkUx8zn-QZ~3lgJtNXV6VPnjUtFPl2MTFKHtvrySYfoJ2oAZ7j5i-O1cSP31Ah6towz6oOktskmM72MvP7M85IvUAEqUdKYFQ3BAlf6mFcvE~lR4eO2yse4IFni79WgYgTyuNwXIXOC8Iz9OvoDuEmD~CCzlWBNJxNdeno3W74fVqxP0xPPZKVer0yJkYdt1sZdscWDKGS9eEZwqbzvGuEQ9dMlfi2e-~AoB4uH6UI-gxTYVzGvkxQIZC068sx0SmWrZRewhdHQAv4YjZFzm-M1MyKNrm3ZoivdFF8GJGMfY9~jsfqpPZd5UUejnj-dKd~-9sse4~wxraA98kTg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":371,"name":"Functional Analysis","url":"https://www.academia.edu/Documents/in/Functional_Analysis"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":267802,"name":"Dimensional","url":"https://www.academia.edu/Documents/in/Dimensional"},{"id":346242,"name":"Hermite Polynomials","url":"https://www.academia.edu/Documents/in/Hermite_Polynomials"},{"id":555663,"name":"Mathematics and Probability","url":"https://www.academia.edu/Documents/in/Mathematics_and_Probability"},{"id":1142720,"name":"Normal Distribution","url":"https://www.academia.edu/Documents/in/Normal_Distribution"},{"id":2902615,"name":"Cartesian product","url":"https://www.academia.edu/Documents/in/Cartesian_product"}],"urls":[{"id":42488751,"url":"https://doi.org/10.1016/j.jfa.2011.09.007"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282240"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282240/On_affinity_relating_two_positive_measures_and_the_connection_coefficients_between_polynomials_orthogonalized_by_these_measures"><img alt="Research paper thumbnail of On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures" class="work-thumbnail" src="https://attachments.academia-assets.com/115484113/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282240/On_affinity_relating_two_positive_measures_and_the_connection_coefficients_between_polynomials_orthogonalized_by_these_measures">On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures</a></div><div class="wp-workCard_item"><span>Applied Mathematics and Computation</span><span>, Feb 1, 2013</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We consider two positive, normalized measures dA (x) and dB (x) related by the relationship dA (x...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We consider two positive, normalized measures dA (x) and dB (x) related by the relationship dA (x) = C x+D dB (x) or by dA (x) = C x 2 +E dB (x) and dB (x) is symmetric. We show that then the polynomial sequences {an (x)} , {bn (x)} orthogonal with respect to these measures are related by the relationship an (x) = bn (x) + κnb n−1 (x) or by an (x) = bn (x) + λnb n−2 (x) for some sequences {κn} and {λn}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {bn (x)} and the sequence {κn} that have a form of Fourier series expansion of the Radon-Nikodym derivative of one measure with respect to the other.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="08ac44f55ab683251c23920a27b6b54b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484113,"asset_id":120282240,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484113/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282240"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282240"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282240; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282240]").text(description); $(".js-view-count[data-work-id=120282240]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282240; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282240']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282240, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "08ac44f55ab683251c23920a27b6b54b" } } $('.js-work-strip[data-work-id=120282240]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282240,"title":"On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We consider two positive, normalized measures dA (x) and dB (x) related by the relationship dA (x) = C x+D dB (x) or by dA (x) = C x 2 +E dB (x) and dB (x) is symmetric. We show that then the polynomial sequences {an (x)} , {bn (x)} orthogonal with respect to these measures are related by the relationship an (x) = bn (x) + κnb n−1 (x) or by an (x) = bn (x) + λnb n−2 (x) for some sequences {κn} and {λn}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {bn (x)} and the sequence {κn} that have a form of Fourier series expansion of the Radon-Nikodym derivative of one measure with respect to the other.","publication_date":{"day":1,"month":2,"year":2013,"errors":{}},"publication_name":"Applied Mathematics and Computation","grobid_abstract_attachment_id":115484113},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282240/On_affinity_relating_two_positive_measures_and_the_connection_coefficients_between_polynomials_orthogonalized_by_these_measures","translated_internal_url":"","created_at":"2024-05-30T11:35:37.072-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484113,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484113/thumbnails/1.jpg","file_name":"1202.pdf","download_url":"https://www.academia.edu/attachments/115484113/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_affinity_relating_two_positive_measur.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484113/1202-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DOn_affinity_relating_two_positive_measur.pdf\u0026Expires=1734522074\u0026Signature=UYCTS8k4RCKZ1ewzuZjY~cnfDOmEN2c5GZ8cHjIMfuloOLInyuvX7sZlWDq-61qFbeQzollMzU3qejFHCcym792uYCwsB4mGa5XlOb64iDhvzauWb7XCl7rMfQZFZhMQKk36fsyYNPzxt1qXwYda8PGLWlzOPg2~Mv955fgqXJGF49G9pE~qltJ6kW3ACHYKGIsbmYOcfuyNLc08W9tG5MrOdwSq7lWqhte2LtzkPwFtmgUUS5xJBRqElhkMsCEciKwgChj09iP8LLm-zWvkIq~zRfnpaC7~G4k-WW~JB4z8WOPNx7O0mC6cMCjrPw4L0wArUVjRN05rllvDtIolHA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_affinity_relating_two_positive_measures_and_the_connection_coefficients_between_polynomials_orthogonalized_by_these_measures","translated_slug":"","page_count":13,"language":"en","content_type":"Work","summary":"We consider two positive, normalized measures dA (x) and dB (x) related by the relationship dA (x) = C x+D dB (x) or by dA (x) = C x 2 +E dB (x) and dB (x) is symmetric. We show that then the polynomial sequences {an (x)} , {bn (x)} orthogonal with respect to these measures are related by the relationship an (x) = bn (x) + κnb n−1 (x) or by an (x) = bn (x) + λnb n−2 (x) for some sequences {κn} and {λn}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {bn (x)} and the sequence {κn} that have a form of Fourier series expansion of the Radon-Nikodym derivative of one measure with respect to the other.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484113,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484113/thumbnails/1.jpg","file_name":"1202.pdf","download_url":"https://www.academia.edu/attachments/115484113/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_affinity_relating_two_positive_measur.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484113/1202-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DOn_affinity_relating_two_positive_measur.pdf\u0026Expires=1734522074\u0026Signature=UYCTS8k4RCKZ1ewzuZjY~cnfDOmEN2c5GZ8cHjIMfuloOLInyuvX7sZlWDq-61qFbeQzollMzU3qejFHCcym792uYCwsB4mGa5XlOb64iDhvzauWb7XCl7rMfQZFZhMQKk36fsyYNPzxt1qXwYda8PGLWlzOPg2~Mv955fgqXJGF49G9pE~qltJ6kW3ACHYKGIsbmYOcfuyNLc08W9tG5MrOdwSq7lWqhte2LtzkPwFtmgUUS5xJBRqElhkMsCEciKwgChj09iP8LLm-zWvkIq~zRfnpaC7~G4k-WW~JB4z8WOPNx7O0mC6cMCjrPw4L0wArUVjRN05rllvDtIolHA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":115484112,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484112/thumbnails/1.jpg","file_name":"1202.pdf","download_url":"https://www.academia.edu/attachments/115484112/download_file","bulk_download_file_name":"On_affinity_relating_two_positive_measur.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484112/1202-libre.pdf?1717097617=\u0026response-content-disposition=attachment%3B+filename%3DOn_affinity_relating_two_positive_measur.pdf\u0026Expires=1734522074\u0026Signature=U~QsemYQtpVbuKDfZijIRKlqk39uw0G-BuAg6XKd62R-wuOLquG-PTVcJoUl0qLTsKnXeM1nZMlmDBs9FrYA0ATHa1Rev-IU18ORROQa5YdNSghy1BBMbYpuloZuwf6g2CDTRNZ4L2ifzAQNzxxP0ehceAMdG8DUS6tsomEq4UxsHAD0KRnHxsBPMMfWjQHGOHMfjQkHmw8jyWkvU6kBD1Lp2nuRyjjaORDyl8yBAwl24A~nCxdWROsseUgzrDsl1KhinzJU3QPQ8ykkWHarJmj6Igstv1Dyc6I1~VppwCrpzmCYPXZsYGqXJxIYLi16cXXju6YI8Kr0B5-yZ2glxg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":12026,"name":"Orthogonal polynomials","url":"https://www.academia.edu/Documents/in/Orthogonal_polynomials"},{"id":556845,"name":"Numerical Analysis and Computational Mathematics","url":"https://www.academia.edu/Documents/in/Numerical_Analysis_and_Computational_Mathematics"},{"id":1228829,"name":"Fourier Series","url":"https://www.academia.edu/Documents/in/Fourier_Series"},{"id":3777178,"name":"Polynomial","url":"https://www.academia.edu/Documents/in/Polynomial"}],"urls":[{"id":42488750,"url":"http://arxiv.org/pdf/1202.1606"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282239"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282239/Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths"><img alt="Research paper thumbnail of Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths" class="work-thumbnail" src="https://attachments.academia-assets.com/115484116/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282239/Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths">Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jun 23, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We are studying stationary random processes with conditional polynomial moments that allow a cont...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification, are important because they are relatively easy to simulate. One does not have to care about the distribution of their jumps which is always difficult to find. Among those processes with the continuous path are the Ornstein-Uhlenbeck process, the Gamma process, the process with Arcsin or Wigner margins and the Theta functions as the transition densities and others. We give a simple criterion for the stationary process to have a continuous path modification expressed in terms of skewness and excess kurtosis of the marginal distribution.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3bafd370e1429d60d119eb0e2fb7bbfc" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484116,"asset_id":120282239,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484116/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282239"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282239"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282239; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282239]").text(description); $(".js-view-count[data-work-id=120282239]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282239; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282239']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282239, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "3bafd370e1429d60d119eb0e2fb7bbfc" } } $('.js-work-strip[data-work-id=120282239]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282239,"title":"Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths","translated_title":"","metadata":{"publisher":"Cornell University","ai_title_tag":"Stationary Markov Processes with Polynomial Moments","grobid_abstract":"We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification, are important because they are relatively easy to simulate. One does not have to care about the distribution of their jumps which is always difficult to find. Among those processes with the continuous path are the Ornstein-Uhlenbeck process, the Gamma process, the process with Arcsin or Wigner margins and the Theta functions as the transition densities and others. We give a simple criterion for the stationary process to have a continuous path modification expressed in terms of skewness and excess kurtosis of the marginal distribution.","publication_date":{"day":23,"month":6,"year":2022,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":115484116},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282239/Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths","translated_internal_url":"","created_at":"2024-05-30T11:35:36.848-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484116,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484116/thumbnails/1.jpg","file_name":"2206.11798.pdf","download_url":"https://www.academia.edu/attachments/115484116/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Stationary_Markov_stochastic_processes_w.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484116/2206.11798-libre.pdf?1717097620=\u0026response-content-disposition=attachment%3B+filename%3DStationary_Markov_stochastic_processes_w.pdf\u0026Expires=1734522074\u0026Signature=LrI~Mzma6NAjRDmAtPFu9GwIJYMtJHW6nP9LgmoaycWM7d8XElIQ-m0DMWwN1vz-urRC0vs8H3K5yp1fy1U4z-32ePYgpwT3IhO2e-3qdqB3u~CNm5~MYnz5hfWzH-jBd6XZvUJRUjfQi0zdq9SSFmqbRQChOcGhWMcd~OpKApXPQtx4E3E4bdixeu9KBTASKmoyEV6TQTYVOeZsY2Txb1EFZkI4pf1A2Dnel4MzfyT6tj33ucLRVZ2ewZ6ntim47hdx8SJB-zeXTJwo3LRg4qCtPxgvKhd~tcRUxaQVcO22Y8c2HVgXxEykhBDl5kiYAC8uSTjQLoBmX7T8RO6xQw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths","translated_slug":"","page_count":20,"language":"en","content_type":"Work","summary":"We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification, are important because they are relatively easy to simulate. One does not have to care about the distribution of their jumps which is always difficult to find. Among those processes with the continuous path are the Ornstein-Uhlenbeck process, the Gamma process, the process with Arcsin or Wigner margins and the Theta functions as the transition densities and others. We give a simple criterion for the stationary process to have a continuous path modification expressed in terms of skewness and excess kurtosis of the marginal distribution.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484116,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484116/thumbnails/1.jpg","file_name":"2206.11798.pdf","download_url":"https://www.academia.edu/attachments/115484116/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Stationary_Markov_stochastic_processes_w.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484116/2206.11798-libre.pdf?1717097620=\u0026response-content-disposition=attachment%3B+filename%3DStationary_Markov_stochastic_processes_w.pdf\u0026Expires=1734522074\u0026Signature=LrI~Mzma6NAjRDmAtPFu9GwIJYMtJHW6nP9LgmoaycWM7d8XElIQ-m0DMWwN1vz-urRC0vs8H3K5yp1fy1U4z-32ePYgpwT3IhO2e-3qdqB3u~CNm5~MYnz5hfWzH-jBd6XZvUJRUjfQi0zdq9SSFmqbRQChOcGhWMcd~OpKApXPQtx4E3E4bdixeu9KBTASKmoyEV6TQTYVOeZsY2Txb1EFZkI4pf1A2Dnel4MzfyT6tj33ucLRVZ2ewZ6ntim47hdx8SJB-zeXTJwo3LRg4qCtPxgvKhd~tcRUxaQVcO22Y8c2HVgXxEykhBDl5kiYAC8uSTjQLoBmX7T8RO6xQw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":16460,"name":"Statistical Physics","url":"https://www.academia.edu/Documents/in/Statistical_Physics"},{"id":191286,"name":"Kurtosis","url":"https://www.academia.edu/Documents/in/Kurtosis"},{"id":480922,"name":"Skewness","url":"https://www.academia.edu/Documents/in/Skewness"},{"id":3777178,"name":"Polynomial","url":"https://www.academia.edu/Documents/in/Polynomial"}],"urls":[{"id":42488749,"url":"https://arxiv.org/pdf/2206.11798"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282238"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282238/Some_Remarks_Concerning_the_Dual_Control_Problem"><img alt="Research paper thumbnail of Some Remarks Concerning the Dual Control Problem" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282238/Some_Remarks_Concerning_the_Dual_Control_Problem">Some Remarks Concerning the Dual Control Problem</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">ABSTRACT L&amp;#39;auteur considere le probleme simultane d&amp;#39;estimation des parametres d&a...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">ABSTRACT L&amp;#39;auteur considere le probleme simultane d&amp;#39;estimation des parametres d&amp;#39;un systeme et du controle optimal de ce meme systeme. Plus precisement on considere un systeme donne par: xsub(i+1)=f(xsubi, u subi, n, p), pinVlhkRsupd avec une fonction de cout de la forme: J(pi )=lim infsub(Narrrinfin) E(sumsub(i=0)sup(N - 1) rho (usubi, x subi)). Dans une premiere partie, l&amp;#39;auteur propose une generalisation de la martingale de Mandl (cas lineaire quadratique) et montre que pour tout suite d&amp;#39;estimateurs (psubi) de (psubi) &amp;quot;stabilisante et reguliere&amp;quot; dans un certain sens on a: limsub(Narrrinfin) (1/N)sumsub(i=0)sup(N - 1) rho (xsubi, u(xsubi, psubi))=J(pi )=infsub pi J(pi ) ou pi =(u(x sub0, psub0), u(xsub1, psub1), ...). Ces resultats sont ensuite appliques au cas lineaire quadratique et permettent de reobtenir les resultats principaux de Mandl. Ensuite est proposee une methode (appelee naturelle) d&amp;#39;estimation de la forme psub(i+1)= pi subi=psubi - mu subiFsub(i+1)(omega , psubi) si pi subiinM ou psub(i+1)= projection de pi subi sur M si pi subi notinM, ou Fsub(i+1)(omega , p) est une suite de vecteurs aleatoires p- dimensionnels, (mu subi) une suite de reels positifs avec mu sub0=1, mu subiin(0, 1) et MlhkRsupp est ferme convexe. Sous certaines conditions de regularite cette procedure d&amp;#39;estimation converge vers p, et de plus pi =(u(xsub0, psub0), u(xsub0, psub1), ...) realise infsub pi J(pi ). Ainsi le probleme d&amp;#39;estimation et de controle du systeme se remene au probleme de la convergence de cette procedure d&amp;#39;estimation. (For the entire collection see MR 80f:90006a.)</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282238"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282238"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282238; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282238]").text(description); $(".js-view-count[data-work-id=120282238]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282238; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282238']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282238, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=120282238]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282238,"title":"Some Remarks Concerning the Dual Control Problem","translated_title":"","metadata":{"abstract":"ABSTRACT L\u0026amp;#39;auteur considere le probleme simultane d\u0026amp;#39;estimation des parametres d\u0026amp;#39;un systeme et du controle optimal de ce meme systeme. Plus precisement on considere un systeme donne par: xsub(i+1)=f(xsubi, u subi, n, p), pinVlhkRsupd avec une fonction de cout de la forme: J(pi )=lim infsub(Narrrinfin) E(sumsub(i=0)sup(N - 1) rho (usubi, x subi)). Dans une premiere partie, l\u0026amp;#39;auteur propose une generalisation de la martingale de Mandl (cas lineaire quadratique) et montre que pour tout suite d\u0026amp;#39;estimateurs (psubi) de (psubi) \u0026amp;quot;stabilisante et reguliere\u0026amp;quot; dans un certain sens on a: limsub(Narrrinfin) (1/N)sumsub(i=0)sup(N - 1) rho (xsubi, u(xsubi, psubi))=J(pi )=infsub pi J(pi ) ou pi =(u(x sub0, psub0), u(xsub1, psub1), ...). Ces resultats sont ensuite appliques au cas lineaire quadratique et permettent de reobtenir les resultats principaux de Mandl. Ensuite est proposee une methode (appelee naturelle) d\u0026amp;#39;estimation de la forme psub(i+1)= pi subi=psubi - mu subiFsub(i+1)(omega , psubi) si pi subiinM ou psub(i+1)= projection de pi subi sur M si pi subi notinM, ou Fsub(i+1)(omega , p) est une suite de vecteurs aleatoires p- dimensionnels, (mu subi) une suite de reels positifs avec mu sub0=1, mu subiin(0, 1) et MlhkRsupp est ferme convexe. Sous certaines conditions de regularite cette procedure d\u0026amp;#39;estimation converge vers p, et de plus pi =(u(xsub0, psub0), u(xsub0, psub1), ...) realise infsub pi J(pi ). Ainsi le probleme d\u0026amp;#39;estimation et de controle du systeme se remene au probleme de la convergence de cette procedure d\u0026amp;#39;estimation. (For the entire collection see MR 80f:90006a.)","publication_date":{"day":17,"month":12,"year":2020,"errors":{}}},"translated_abstract":"ABSTRACT L\u0026amp;#39;auteur considere le probleme simultane d\u0026amp;#39;estimation des parametres d\u0026amp;#39;un systeme et du controle optimal de ce meme systeme. Plus precisement on considere un systeme donne par: xsub(i+1)=f(xsubi, u subi, n, p), pinVlhkRsupd avec une fonction de cout de la forme: J(pi )=lim infsub(Narrrinfin) E(sumsub(i=0)sup(N - 1) rho (usubi, x subi)). Dans une premiere partie, l\u0026amp;#39;auteur propose une generalisation de la martingale de Mandl (cas lineaire quadratique) et montre que pour tout suite d\u0026amp;#39;estimateurs (psubi) de (psubi) \u0026amp;quot;stabilisante et reguliere\u0026amp;quot; dans un certain sens on a: limsub(Narrrinfin) (1/N)sumsub(i=0)sup(N - 1) rho (xsubi, u(xsubi, psubi))=J(pi )=infsub pi J(pi ) ou pi =(u(x sub0, psub0), u(xsub1, psub1), ...). Ces resultats sont ensuite appliques au cas lineaire quadratique et permettent de reobtenir les resultats principaux de Mandl. Ensuite est proposee une methode (appelee naturelle) d\u0026amp;#39;estimation de la forme psub(i+1)= pi subi=psubi - mu subiFsub(i+1)(omega , psubi) si pi subiinM ou psub(i+1)= projection de pi subi sur M si pi subi notinM, ou Fsub(i+1)(omega , p) est une suite de vecteurs aleatoires p- dimensionnels, (mu subi) une suite de reels positifs avec mu sub0=1, mu subiin(0, 1) et MlhkRsupp est ferme convexe. Sous certaines conditions de regularite cette procedure d\u0026amp;#39;estimation converge vers p, et de plus pi =(u(xsub0, psub0), u(xsub0, psub1), ...) realise infsub pi J(pi ). Ainsi le probleme d\u0026amp;#39;estimation et de controle du systeme se remene au probleme de la convergence de cette procedure d\u0026amp;#39;estimation. (For the entire collection see MR 80f:90006a.)","internal_url":"https://www.academia.edu/120282238/Some_Remarks_Concerning_the_Dual_Control_Problem","translated_internal_url":"","created_at":"2024-05-30T11:35:36.640-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Some_Remarks_Concerning_the_Dual_Control_Problem","translated_slug":"","page_count":null,"language":"fr","content_type":"Work","summary":"ABSTRACT L\u0026amp;#39;auteur considere le probleme simultane d\u0026amp;#39;estimation des parametres d\u0026amp;#39;un systeme et du controle optimal de ce meme systeme. Plus precisement on considere un systeme donne par: xsub(i+1)=f(xsubi, u subi, n, p), pinVlhkRsupd avec une fonction de cout de la forme: J(pi )=lim infsub(Narrrinfin) E(sumsub(i=0)sup(N - 1) rho (usubi, x subi)). Dans une premiere partie, l\u0026amp;#39;auteur propose une generalisation de la martingale de Mandl (cas lineaire quadratique) et montre que pour tout suite d\u0026amp;#39;estimateurs (psubi) de (psubi) \u0026amp;quot;stabilisante et reguliere\u0026amp;quot; dans un certain sens on a: limsub(Narrrinfin) (1/N)sumsub(i=0)sup(N - 1) rho (xsubi, u(xsubi, psubi))=J(pi )=infsub pi J(pi ) ou pi =(u(x sub0, psub0), u(xsub1, psub1), ...). Ces resultats sont ensuite appliques au cas lineaire quadratique et permettent de reobtenir les resultats principaux de Mandl. Ensuite est proposee une methode (appelee naturelle) d\u0026amp;#39;estimation de la forme psub(i+1)= pi subi=psubi - mu subiFsub(i+1)(omega , psubi) si pi subiinM ou psub(i+1)= projection de pi subi sur M si pi subi notinM, ou Fsub(i+1)(omega , p) est une suite de vecteurs aleatoires p- dimensionnels, (mu subi) une suite de reels positifs avec mu sub0=1, mu subiin(0, 1) et MlhkRsupp est ferme convexe. Sous certaines conditions de regularite cette procedure d\u0026amp;#39;estimation converge vers p, et de plus pi =(u(xsub0, psub0), u(xsub0, psub1), ...) realise infsub pi J(pi ). Ainsi le probleme d\u0026amp;#39;estimation et de controle du systeme se remene au probleme de la convergence de cette procedure d\u0026amp;#39;estimation. (For the entire collection see MR 80f:90006a.)","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[],"research_interests":[{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":3273604,"name":"Dual (grammatical number)","url":"https://www.academia.edu/Documents/in/Dual_grammatical_number_"}],"urls":[{"id":42488748,"url":"https://doi.org/10.1201/9781003071877-10"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282237"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282237/On_connection_between_values_of_Riemann_zeta_function_at_rationals_and_generalized_harmonic_numbers"><img alt="Research paper thumbnail of On connection between values of Riemann zeta function at rationals and generalized harmonic numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/115484121/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282237/On_connection_between_values_of_Riemann_zeta_function_at_rationals_and_generalized_harmonic_numbers">On connection between values of Riemann zeta function at rationals and generalized harmonic numbers</a></div><div class="wp-workCard_item"><span>Mathematica Eterna</span><span>, Jan 12, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer and rational values of arguments to certain rapidly converging series, where some generalized harmonic numbers appear. Most of the results of the paper can be derived from the recent, more advanced results, on the properties of Arakawa-Kaneko zeta functions. We derive our results directly, by solving simple recursions. The form of mentioned above generalized harmonic numbers carries information, about the values of the arguments of Hurwitz function. In particular we prove: 8k 2 N : (k; 1) = (k) = 2 k 1 2 k 1 1</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fddd9654b255cce3acc47ec8705741be" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484121,"asset_id":120282237,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484121/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282237"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282237"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282237; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282237]").text(description); $(".js-view-count[data-work-id=120282237]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282237; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282237']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282237, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "fddd9654b255cce3acc47ec8705741be" } } $('.js-work-strip[data-work-id=120282237]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282237,"title":"On connection between values of Riemann zeta function at rationals and generalized harmonic numbers","translated_title":"","metadata":{"grobid_abstract":"Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer and rational values of arguments to certain rapidly converging series, where some generalized harmonic numbers appear. Most of the results of the paper can be derived from the recent, more advanced results, on the properties of Arakawa-Kaneko zeta functions. We derive our results directly, by solving simple recursions. The form of mentioned above generalized harmonic numbers carries information, about the values of the arguments of Hurwitz function. In particular we prove: 8k 2 N : (k; 1) = (k) = 2 k 1 2 k 1 1","publication_date":{"day":12,"month":1,"year":2015,"errors":{}},"publication_name":"Mathematica Eterna","grobid_abstract_attachment_id":115484121},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282237/On_connection_between_values_of_Riemann_zeta_function_at_rationals_and_generalized_harmonic_numbers","translated_internal_url":"","created_at":"2024-05-30T11:35:36.457-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484121,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484121/thumbnails/1.jpg","file_name":"MA5_4_5.pdf","download_url":"https://www.academia.edu/attachments/115484121/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_connection_between_values_of_Riemann.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484121/MA5_4_5-libre.pdf?1717097611=\u0026response-content-disposition=attachment%3B+filename%3DOn_connection_between_values_of_Riemann.pdf\u0026Expires=1734522074\u0026Signature=ad5iZP0vFj-t1pH3Sg3Qr7jkTqAhYp5wOjzfKv9KHKEKMChpIhaZQoICUayIx4lkCHbE3tAsjTgHgmH9l8HTCotRU3ULygRAAOldLlmTLdCjAutKg4G1uRvT~CvTkTmrkhWk1tMHBOsd956Ghbb-HuJJy5pQi3sQw0JApuZnfMLx8o2ctIkXTyJgc3XmPoddilEkfeV5WZ7z3Oi06IKH0fosPgLa4juLUC76Me6fP1EYJ4KJq5v1EQQW0Wc5E7I6gRGOm7BeqG6QBl6qnP-MVkVbdMR1NelMXDcHfQCKe2ZTOzjLGVx0KQufFx2kseefPE1arnfvsTwFyP0rCpVilw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_connection_between_values_of_Riemann_zeta_function_at_rationals_and_generalized_harmonic_numbers","translated_slug":"","page_count":10,"language":"en","content_type":"Work","summary":"Using Euler transformation of series, we relate values of Hurwitz zeta function (s; t) at integer and rational values of arguments to certain rapidly converging series, where some generalized harmonic numbers appear. Most of the results of the paper can be derived from the recent, more advanced results, on the properties of Arakawa-Kaneko zeta functions. We derive our results directly, by solving simple recursions. The form of mentioned above generalized harmonic numbers carries information, about the values of the arguments of Hurwitz function. In particular we prove: 8k 2 N : (k; 1) = (k) = 2 k 1 2 k 1 1","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484121,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484121/thumbnails/1.jpg","file_name":"MA5_4_5.pdf","download_url":"https://www.academia.edu/attachments/115484121/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_connection_between_values_of_Riemann.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484121/MA5_4_5-libre.pdf?1717097611=\u0026response-content-disposition=attachment%3B+filename%3DOn_connection_between_values_of_Riemann.pdf\u0026Expires=1734522074\u0026Signature=ad5iZP0vFj-t1pH3Sg3Qr7jkTqAhYp5wOjzfKv9KHKEKMChpIhaZQoICUayIx4lkCHbE3tAsjTgHgmH9l8HTCotRU3ULygRAAOldLlmTLdCjAutKg4G1uRvT~CvTkTmrkhWk1tMHBOsd956Ghbb-HuJJy5pQi3sQw0JApuZnfMLx8o2ctIkXTyJgc3XmPoddilEkfeV5WZ7z3Oi06IKH0fosPgLa4juLUC76Me6fP1EYJ4KJq5v1EQQW0Wc5E7I6gRGOm7BeqG6QBl6qnP-MVkVbdMR1NelMXDcHfQCKe2ZTOzjLGVx0KQufFx2kseefPE1arnfvsTwFyP0rCpVilw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":229028,"name":"riemann Hypothesis","url":"https://www.academia.edu/Documents/in/riemann_Hypothesis"},{"id":1494623,"name":"Real Number","url":"https://www.academia.edu/Documents/in/Real_Number"}],"urls":[{"id":42488747,"url":"https://www.longdom.org/articles/on-connection-between-values-of-riemann-zeta-function-at-rationals-and-generalized-harmonic-numbers.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282236"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes"><img alt="Research paper thumbnail of q-Wiener and related processes. A continuous time generalization of Bryc processes" class="work-thumbnail" src="https://attachments.academia-assets.com/115484111/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes">q-Wiener and related processes. A continuous time generalization of Bryc processes</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jul 14, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We collect, scattered through literature, as well as we prove some new properties of two Markov p...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X = (Xt) t≥0 called q−Wiener) depends on one parameter q ∈ (−1, 1] and of the second one (say Y = (Yt) t∈R called (α, q)− Ornstein-Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles Wiener process in the sense that for q = 1 it is Wiener process but also that for |q| < 1 and ∀n ≥ 1 : t n/2 Hn Xt/ √ t|q , where (Hn) n≥0 are the so called q−Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein-Uhlenbeck process. For q = 1 it is a classical OU process. For |q| < 1 it is also stationary with correlation function equal to exp(−α|t − s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a71bb2cf2f7c2e07ecb0b5d2621ce6cb" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484111,"asset_id":120282236,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484111/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282236"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282236"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282236; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282236]").text(description); $(".js-view-count[data-work-id=120282236]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282236; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282236']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282236, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a71bb2cf2f7c2e07ecb0b5d2621ce6cb" } } $('.js-work-strip[data-work-id=120282236]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282236,"title":"q-Wiener and related processes. A continuous time generalization of Bryc processes","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X = (Xt) t≥0 called q−Wiener) depends on one parameter q ∈ (−1, 1] and of the second one (say Y = (Yt) t∈R called (α, q)− Ornstein-Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles Wiener process in the sense that for q = 1 it is Wiener process but also that for |q| \u003c 1 and ∀n ≥ 1 : t n/2 Hn Xt/ √ t|q , where (Hn) n≥0 are the so called q−Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein-Uhlenbeck process. For q = 1 it is a classical OU process. For |q| \u003c 1 it is also stationary with correlation function equal to exp(−α|t − s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.","publication_date":{"day":14,"month":7,"year":2005,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":115484111},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes","translated_internal_url":"","created_at":"2024-05-30T11:35:36.268-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484111,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484111/thumbnails/1.jpg","file_name":"0507303.pdf","download_url":"https://www.academia.edu/attachments/115484111/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"q_Wiener_and_related_processes_A_continu.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484111/0507303-libre.pdf?1717097627=\u0026response-content-disposition=attachment%3B+filename%3Dq_Wiener_and_related_processes_A_continu.pdf\u0026Expires=1734522074\u0026Signature=cFJgekt1Bz0gVAbes-PLPkFHsrXrNivfKBetwP86Dmp4nSrHl5ImHOBTDv5PSD3ctL-oXs593FEiQ~5c9mCjAITw0XQBJwsgBO7bWyigFAEiu9YYTNKWyop60F5ryyELIMMVfXrMTHd-7G-3UPca-P0y6LAhvhU4kifq-9i1ixnnUeXbBViogGWHxmdBuyTcz~PXiK-hfC7neNpCX8jdoLp41PmVT6d5ivW11r8efZgoZF6CW2mKg8GgWKPuWGzxvBKCk1ceSyj4rY8m7oit~3budDEdC6JHqsTauLa4GyhpIyoJ2TL6QQJXUCMqnh1e7SNTbTmGORlnPnZ966IYwg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes","translated_slug":"","page_count":25,"language":"en","content_type":"Work","summary":"We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X = (Xt) t≥0 called q−Wiener) depends on one parameter q ∈ (−1, 1] and of the second one (say Y = (Yt) t∈R called (α, q)− Ornstein-Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles Wiener process in the sense that for q = 1 it is Wiener process but also that for |q| \u003c 1 and ∀n ≥ 1 : t n/2 Hn Xt/ √ t|q , where (Hn) n≥0 are the so called q−Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein-Uhlenbeck process. For q = 1 it is a classical OU process. For |q| \u003c 1 it is also stationary with correlation function equal to exp(−α|t − s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484111,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484111/thumbnails/1.jpg","file_name":"0507303.pdf","download_url":"https://www.academia.edu/attachments/115484111/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"q_Wiener_and_related_processes_A_continu.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484111/0507303-libre.pdf?1717097627=\u0026response-content-disposition=attachment%3B+filename%3Dq_Wiener_and_related_processes_A_continu.pdf\u0026Expires=1734522074\u0026Signature=cFJgekt1Bz0gVAbes-PLPkFHsrXrNivfKBetwP86Dmp4nSrHl5ImHOBTDv5PSD3ctL-oXs593FEiQ~5c9mCjAITw0XQBJwsgBO7bWyigFAEiu9YYTNKWyop60F5ryyELIMMVfXrMTHd-7G-3UPca-P0y6LAhvhU4kifq-9i1ixnnUeXbBViogGWHxmdBuyTcz~PXiK-hfC7neNpCX8jdoLp41PmVT6d5ivW11r8efZgoZF6CW2mKg8GgWKPuWGzxvBKCk1ceSyj4rY8m7oit~3budDEdC6JHqsTauLa4GyhpIyoJ2TL6QQJXUCMqnh1e7SNTbTmGORlnPnZ966IYwg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":115484110,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484110/thumbnails/1.jpg","file_name":"0507303.pdf","download_url":"https://www.academia.edu/attachments/115484110/download_file","bulk_download_file_name":"q_Wiener_and_related_processes_A_continu.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484110/0507303-libre.pdf?1717097623=\u0026response-content-disposition=attachment%3B+filename%3Dq_Wiener_and_related_processes_A_continu.pdf\u0026Expires=1734522074\u0026Signature=ar-MChd~HMnik2pkRIkrW9jYJvbxkok-EZdaBzRTtx2MjvZMKtDLJx48mt04xxQMjYQXdwoGXKLAQg6QOpOJ8e-Ax9B3QLbRKiZb84U8GB708hCRLc4AaovmIeR2laQUIbC~BSY3cn~7OcPElivlIsGkATbXiblFtvJOc~hizFZZDQhl0py8aJC0sGOBfmsoTQdqsdVr9sLM-WhX9JWRD-nzBOpiBt3t08Fcl201kmWQYFfDflx3Cvgk-bL9uDB4sn1OMn6ujZD~88eh4w6u1fXqqSMCBP43REXqBKS64aztz7bs7jNJ~dDe1Y5tCG14JH4AR9Z-EOZVzLk4NtG-Wg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":389829,"name":"Gaussian distribution","url":"https://www.academia.edu/Documents/in/Gaussian_distribution"},{"id":679783,"name":"Boolean Satisfiability","url":"https://www.academia.edu/Documents/in/Boolean_Satisfiability"},{"id":741671,"name":"Markov Process","url":"https://www.academia.edu/Documents/in/Markov_Process"},{"id":2382100,"name":"Correlation function","url":"https://www.academia.edu/Documents/in/Correlation_function"},{"id":3115633,"name":"Discrete time","url":"https://www.academia.edu/Documents/in/Discrete_time"}],"urls":[{"id":42488746,"url":"http://arxiv.org/pdf/math/0507303"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282234"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282234/On_probabilistic_aspects_of_Chebyshev_polynomials"><img alt="Research paper thumbnail of On probabilistic aspects of Chebyshev polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/115484108/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282234/On_probabilistic_aspects_of_Chebyshev_polynomials">On probabilistic aspects of Chebyshev polynomials</a></div><div class="wp-workCard_item"><span>Statistics & Probability Letters</span><span>, Feb 1, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We examine the properties of distributions with the density of the form: 2Anc n−2 √ c 2 −x 2 π n ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We examine the properties of distributions with the density of the form: 2Anc n−2 √ c 2 −x 2 π n j=1 (c(1+a 2 j)−2a j x) , where c, a 1 ,. .. , an are some parameters and An a suitable constant. We find general forms of An, of k−th moment and of k−th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey-Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a 1 ,. .. , an forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6. n j=1 (c(1 + a 2 j) − 2a j x) , defined for n ≥ 0, |x| ≤ c, with c > 0, |a j | < 1, j = 1,. .. , n. Here A n is a normalizing constant being the function of parameters a 1 ,. .. , a n. We will call this family generalized Kesten-McKay distributions. The name is justified by the fact that the distribution with the following density: (1.2) f KMK2 (x|2/a, a, −a) = v 4(v − 1) − x 2 2π(v 2 − x 2) , where v = 1+1/a 2 and |a| < 1 has been defined, described and what is more, derived in [5] or [9]. Then the name Kesten-McKay distribution has been attributed to this distribution in the literature that appeared after 1981. Thus, it is justified to call the distribution defined by (1.1) a generalized Kesten-McKay (GKM) distribution. Note also that for n = 0 the distribution with the density f KMK0 (x|c) becomes Wigner or semicircle distribution with parameter c.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3decd0381c45891822fdf4908c89aed0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484108,"asset_id":120282234,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484108/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282234"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282234"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282234; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282234]").text(description); $(".js-view-count[data-work-id=120282234]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282234; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282234']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282234, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "3decd0381c45891822fdf4908c89aed0" } } $('.js-work-strip[data-work-id=120282234]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282234,"title":"On probabilistic aspects of Chebyshev polynomials","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We examine the properties of distributions with the density of the form: 2Anc n−2 √ c 2 −x 2 π n j=1 (c(1+a 2 j)−2a j x) , where c, a 1 ,. .. , an are some parameters and An a suitable constant. We find general forms of An, of k−th moment and of k−th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey-Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a 1 ,. .. , an forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6. n j=1 (c(1 + a 2 j) − 2a j x) , defined for n ≥ 0, |x| ≤ c, with c \u003e 0, |a j | \u003c 1, j = 1,. .. , n. Here A n is a normalizing constant being the function of parameters a 1 ,. .. , a n. We will call this family generalized Kesten-McKay distributions. The name is justified by the fact that the distribution with the following density: (1.2) f KMK2 (x|2/a, a, −a) = v 4(v − 1) − x 2 2π(v 2 − x 2) , where v = 1+1/a 2 and |a| \u003c 1 has been defined, described and what is more, derived in [5] or [9]. Then the name Kesten-McKay distribution has been attributed to this distribution in the literature that appeared after 1981. Thus, it is justified to call the distribution defined by (1.1) a generalized Kesten-McKay (GKM) distribution. Note also that for n = 0 the distribution with the density f KMK0 (x|c) becomes Wigner or semicircle distribution with parameter c.","publication_date":{"day":1,"month":2,"year":2019,"errors":{}},"publication_name":"Statistics \u0026 Probability Letters","grobid_abstract_attachment_id":115484108},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282234/On_probabilistic_aspects_of_Chebyshev_polynomials","translated_internal_url":"","created_at":"2024-05-30T11:35:34.944-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484108,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484108/thumbnails/1.jpg","file_name":"1507.pdf","download_url":"https://www.academia.edu/attachments/115484108/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_probabilistic_aspects_of_Chebyshev_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484108/1507-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DOn_probabilistic_aspects_of_Chebyshev_po.pdf\u0026Expires=1734522074\u0026Signature=fkU8SF9oX4lUXQaSeCcs0gJEomCmpXPl8PpeKN0eroslHWdDagjkAeEXv7f891l8P73fz324TaluqgY-uxclxDUrtfndEj1OV5M6PQ3Xvpzc951rb2XV-qP6~78wBXOGn5p-iqTvtfEHfO8WO-oNhJonV6Xp~FxNn9hqMvqIzKMuO95kyTDrV-qhiQ-OjX2nCEaheZQf4HSdzyjHWAneQTf95a64hM2awchjNXKyqemjTJ0DbByXM25aCNRxsovz9Ge-1-XZL5Lv1IbIUQ~Xmu9DVJR-ISWDpk0JAkOkgRHf7kVICXi4k6E-QOdMkUpST96ZjQZrSQYo0Ulscqnexw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_probabilistic_aspects_of_Chebyshev_polynomials","translated_slug":"","page_count":13,"language":"en","content_type":"Work","summary":"We examine the properties of distributions with the density of the form: 2Anc n−2 √ c 2 −x 2 π n j=1 (c(1+a 2 j)−2a j x) , where c, a 1 ,. .. , an are some parameters and An a suitable constant. We find general forms of An, of k−th moment and of k−th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey-Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a 1 ,. .. , an forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6. n j=1 (c(1 + a 2 j) − 2a j x) , defined for n ≥ 0, |x| ≤ c, with c \u003e 0, |a j | \u003c 1, j = 1,. .. , n. Here A n is a normalizing constant being the function of parameters a 1 ,. .. , a n. We will call this family generalized Kesten-McKay distributions. The name is justified by the fact that the distribution with the following density: (1.2) f KMK2 (x|2/a, a, −a) = v 4(v − 1) − x 2 2π(v 2 − x 2) , where v = 1+1/a 2 and |a| \u003c 1 has been defined, described and what is more, derived in [5] or [9]. Then the name Kesten-McKay distribution has been attributed to this distribution in the literature that appeared after 1981. Thus, it is justified to call the distribution defined by (1.1) a generalized Kesten-McKay (GKM) distribution. Note also that for n = 0 the distribution with the density f KMK0 (x|c) becomes Wigner or semicircle distribution with parameter c.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484108,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484108/thumbnails/1.jpg","file_name":"1507.pdf","download_url":"https://www.academia.edu/attachments/115484108/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_probabilistic_aspects_of_Chebyshev_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484108/1507-libre.pdf?1717097618=\u0026response-content-disposition=attachment%3B+filename%3DOn_probabilistic_aspects_of_Chebyshev_po.pdf\u0026Expires=1734522074\u0026Signature=fkU8SF9oX4lUXQaSeCcs0gJEomCmpXPl8PpeKN0eroslHWdDagjkAeEXv7f891l8P73fz324TaluqgY-uxclxDUrtfndEj1OV5M6PQ3Xvpzc951rb2XV-qP6~78wBXOGn5p-iqTvtfEHfO8WO-oNhJonV6Xp~FxNn9hqMvqIzKMuO95kyTDrV-qhiQ-OjX2nCEaheZQf4HSdzyjHWAneQTf95a64hM2awchjNXKyqemjTJ0DbByXM25aCNRxsovz9Ge-1-XZL5Lv1IbIUQ~Xmu9DVJR-ISWDpk0JAkOkgRHf7kVICXi4k6E-QOdMkUpST96ZjQZrSQYo0Ulscqnexw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":115484109,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484109/thumbnails/1.jpg","file_name":"1507.pdf","download_url":"https://www.academia.edu/attachments/115484109/download_file","bulk_download_file_name":"On_probabilistic_aspects_of_Chebyshev_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484109/1507-libre.pdf?1717097619=\u0026response-content-disposition=attachment%3B+filename%3DOn_probabilistic_aspects_of_Chebyshev_po.pdf\u0026Expires=1734522074\u0026Signature=EsiFNiputI8ee4QzHgP9btvE~bWfIvXgHk0BU53dEJP2x1KDUytqXYhQ7-sJ~fjFOZNKus2oFnr-I-Kn12kMt-sDuO7gEjeeqNx5PfiXZnrXynpEW-DGIhelfslrsMy7IfVSExoFxf4o1G0OZ7NRq51r5YAuUYg5KHBXrSfq8kE4rwRknnTlvqjqION-IJkHyjFh8hVBpzKxMeKGgAfaojxHnMgvQFeYxhTfpdOucAgB0HSZq7MKDQA93BKm5tW3Jbsd4joHnNLkLmqtNe18QDqUsBLRHkJtvrG3d6nQ6~vUcRQ2KBN4XYNh3KcRVUiyZqnv6Z5DJwap8kyf0i~uMQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":747,"name":"Econometrics","url":"https://www.academia.edu/Documents/in/Econometrics"},{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"},{"id":12026,"name":"Orthogonal polynomials","url":"https://www.academia.edu/Documents/in/Orthogonal_polynomials"},{"id":2534646,"name":"Chebyshev polynomials","url":"https://www.academia.edu/Documents/in/Chebyshev_polynomials"},{"id":2730277,"name":"Chebyshev filter","url":"https://www.academia.edu/Documents/in/Chebyshev_filter"}],"urls":[{"id":42488745,"url":"http://arxiv.org/pdf/1507.03191"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282233"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282233/On_three_dimensional_multivariate_version_of_q_Normal_distribution_and_probabilistic_interpretations_of_Askey_Wilson_Al_Salam_Chihara_and_q_ultraspherical_polynomials"><img alt="Research paper thumbnail of On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey–Wilson, Al-Salam–Chihara and q-ultraspherical polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/115484122/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282233/On_three_dimensional_multivariate_version_of_q_Normal_distribution_and_probabilistic_interpretations_of_Askey_Wilson_Al_Salam_Chihara_and_q_ultraspherical_polynomials">On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey–Wilson, Al-Salam–Chihara and q-ultraspherical polynomials</a></div><div class="wp-workCard_item"><span>Journal of Mathematical Analysis and Applications</span><span>, Jun 1, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study properties of compactly supported, 4 parameter (ρ 12 , ρ 23 , ρ 13 , q) ∈ (−1, 1) ×4 fam...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We study properties of compactly supported, 4 parameter (ρ 12 , ρ 23 , ρ 13 , q) ∈ (−1, 1) ×4 family of continuous type 3 dimensional distributions, that have the property that for q → 1 − this family tends to some 3 dimensional Normal distribution. For q = 0 we deal with 3 dimensional generalization of Kesten-McKay distribution. In a very special case when ρ 12 ρ 13 ρ 23 = q all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order n and are polynomials of the same order n in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey-Wilson, Al-Salam-Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="05f38fdda0ae638849d9544c63295f44" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484122,"asset_id":120282233,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484122/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282233"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282233"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282233; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282233]").text(description); $(".js-view-count[data-work-id=120282233]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282233; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282233']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282233, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "05f38fdda0ae638849d9544c63295f44" } } $('.js-work-strip[data-work-id=120282233]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282233,"title":"On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey–Wilson, Al-Salam–Chihara and q-ultraspherical polynomials","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We study properties of compactly supported, 4 parameter (ρ 12 , ρ 23 , ρ 13 , q) ∈ (−1, 1) ×4 family of continuous type 3 dimensional distributions, that have the property that for q → 1 − this family tends to some 3 dimensional Normal distribution. For q = 0 we deal with 3 dimensional generalization of Kesten-McKay distribution. In a very special case when ρ 12 ρ 13 ρ 23 = q all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order n and are polynomials of the same order n in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey-Wilson, Al-Salam-Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials.","publication_date":{"day":1,"month":6,"year":2019,"errors":{}},"publication_name":"Journal of Mathematical Analysis and Applications","grobid_abstract_attachment_id":115484122},"translated_abstract":null,"internal_url":"https://www.academia.edu/120282233/On_three_dimensional_multivariate_version_of_q_Normal_distribution_and_probabilistic_interpretations_of_Askey_Wilson_Al_Salam_Chihara_and_q_ultraspherical_polynomials","translated_internal_url":"","created_at":"2024-05-30T11:35:34.744-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":115484122,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484122/thumbnails/1.jpg","file_name":"1712.pdf","download_url":"https://www.academia.edu/attachments/115484122/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_three_dimensional_multivariate_versio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484122/1712-libre.pdf?1717097617=\u0026response-content-disposition=attachment%3B+filename%3DOn_three_dimensional_multivariate_versio.pdf\u0026Expires=1734522074\u0026Signature=dIetTf7MBAUbpbPr9IJHIb~LfmMucc~lQd2vwRTUBzAzfHZgyYtJ83hsp-0zg7-0vwp2HspxocdI~cONIvciaZpmjDCkhcDt075EYsJLGQgE0d9Asm6dOgmO166k~mrTF8YviclqFXzRshS84BYV9BZTiEBYEmhweQBBcoLzSbPh~v05lGOTVRIkq5i8AuCQ4G3uUZ2u-SCvE8ctJj~I3x8K7Q3UCSJ~mCyVM06ij6BdpiIV-lLD8tEKU4ghXwffIgY9aEawWowRs8FUku~oknNARxtGwkCJgDBZc7bkTtQyUUZt2wi-sI31hV6ZIyapR-cLM-PwEQ6Z1y-~y1PTqA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_three_dimensional_multivariate_version_of_q_Normal_distribution_and_probabilistic_interpretations_of_Askey_Wilson_Al_Salam_Chihara_and_q_ultraspherical_polynomials","translated_slug":"","page_count":15,"language":"en","content_type":"Work","summary":"We study properties of compactly supported, 4 parameter (ρ 12 , ρ 23 , ρ 13 , q) ∈ (−1, 1) ×4 family of continuous type 3 dimensional distributions, that have the property that for q → 1 − this family tends to some 3 dimensional Normal distribution. For q = 0 we deal with 3 dimensional generalization of Kesten-McKay distribution. In a very special case when ρ 12 ρ 13 ρ 23 = q all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order n and are polynomials of the same order n in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey-Wilson, Al-Salam-Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484122,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484122/thumbnails/1.jpg","file_name":"1712.pdf","download_url":"https://www.academia.edu/attachments/115484122/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_three_dimensional_multivariate_versio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484122/1712-libre.pdf?1717097617=\u0026response-content-disposition=attachment%3B+filename%3DOn_three_dimensional_multivariate_versio.pdf\u0026Expires=1734522074\u0026Signature=dIetTf7MBAUbpbPr9IJHIb~LfmMucc~lQd2vwRTUBzAzfHZgyYtJ83hsp-0zg7-0vwp2HspxocdI~cONIvciaZpmjDCkhcDt075EYsJLGQgE0d9Asm6dOgmO166k~mrTF8YviclqFXzRshS84BYV9BZTiEBYEmhweQBBcoLzSbPh~v05lGOTVRIkq5i8AuCQ4G3uUZ2u-SCvE8ctJj~I3x8K7Q3UCSJ~mCyVM06ij6BdpiIV-lLD8tEKU4ghXwffIgY9aEawWowRs8FUku~oknNARxtGwkCJgDBZc7bkTtQyUUZt2wi-sI31hV6ZIyapR-cLM-PwEQ6Z1y-~y1PTqA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":12026,"name":"Orthogonal polynomials","url":"https://www.academia.edu/Documents/in/Orthogonal_polynomials"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":346242,"name":"Hermite Polynomials","url":"https://www.academia.edu/Documents/in/Hermite_Polynomials"},{"id":518958,"name":"Mathematical Analysis and Applications","url":"https://www.academia.edu/Documents/in/Mathematical_Analysis_and_Applications"},{"id":1237788,"name":"Electrical And Electronic Engineering","url":"https://www.academia.edu/Documents/in/Electrical_And_Electronic_Engineering"}],"urls":[{"id":42488744,"url":"https://doi.org/10.1016/j.jmaa.2019.02.002"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282232"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282232/Elliptically_contoured_random_variables_and_their_application_to_the_extension_of_the_Kalman_filter"><img alt="Research paper thumbnail of Elliptically contoured random variables and their application to the extension of the Kalman filter" class="work-thumbnail" src="https://attachments.academia-assets.com/115484124/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282232/Elliptically_contoured_random_variables_and_their_application_to_the_extension_of_the_Kalman_filter">Elliptically contoured random variables and their application to the extension of the Kalman filter</a></div><div class="wp-workCard_item"><span>Computers & mathematics with applications</span><span>, 1990</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We summarize some properties of elhptlcall} contoured (e c.) random xanables and indicate their p...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We summarize some properties of elhptlcall} contoured (e c.) random xanables and indicate their possible apphcation in engineering practice. In parttcular after presenting some nex~ results ~e appl} them to modify the Kalman filter. The modified ',erslon is robust against the extenston of the no~ses' model from normal to ellipttcall~ contoured ,%'ole on notation All vector are columns x T means transposition of xector .v lhence v T is a row vector). Matrices and vectors are multiplied according to usual rules. If</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="44a8157f511a27f0c656f8a558559ec2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484124,"asset_id":120282232,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484124/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282232"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282232"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282232; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282232]").text(description); $(".js-view-count[data-work-id=120282232]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282232; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282232']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282232, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "44a8157f511a27f0c656f8a558559ec2" } } $('.js-work-strip[data-work-id=120282232]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282232,"title":"Elliptically contoured random variables and their application to the extension of the Kalman filter","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We summarize some properties of elhptlcall} contoured (e c.) random xanables and indicate their possible apphcation in engineering practice. In parttcular after presenting some nex~ results ~e appl} them to modify the Kalman filter. The modified ',erslon is robust against the extenston of the no~ses' model from normal to ellipttcall~ contoured ,%'ole on notation All vector are columns x T means transposition of xector .v lhence v T is a row vector). Matrices and vectors are multiplied according to usual rules. 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If","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484124,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484124/thumbnails/1.jpg","file_name":"81946991.pdf","download_url":"https://www.academia.edu/attachments/115484124/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Elliptically_contoured_random_variables.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484124/81946991-libre.pdf?1717097613=\u0026response-content-disposition=attachment%3B+filename%3DElliptically_contoured_random_variables.pdf\u0026Expires=1734522074\u0026Signature=ZIUWWmKrfo~iK5IVfvjDnMu-uRZFuP39gQ50WKc8Hx5uTojXnbKVqiT0r5hSm23Jqv3AGCoQszfmxTRsfbSCMkToRZd8AOKPdAabgFbDDu1f-cWQyVwu~ZDAbEnZncbnzKy7uIa4wSMEPeDtbP3HBx~t6mMdf6Ju0ylucekg7T1GLAHuGWj-iYbnGRO7xdmo1UhINpwVDwSQQjhHGyT5~R9zKX6gi61kcN0tj8Ndzsmdap1s35X7MHuJW3SdEaupFMef9bGlGOZCc0axanWNl2dfqI-gEK6gM2I9-C-le6mFyodWVBAENNgrAzK4KiX7~GtC4vB5tH84DXE8eKzp5g__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":49146,"name":"Kalman Filter","url":"https://www.academia.edu/Documents/in/Kalman_Filter"},{"id":80414,"name":"Mathematical Sciences","url":"https://www.academia.edu/Documents/in/Mathematical_Sciences"},{"id":171894,"name":"Computers and Mathematics with Applications 59 (2010) 35783582","url":"https://www.academia.edu/Documents/in/Computers_and_Mathematics_with_Applications_59_2010_35783582"},{"id":611819,"name":"Discrete random variable","url":"https://www.academia.edu/Documents/in/Discrete_random_variable"}],"urls":[{"id":42488743,"url":"https://doi.org/10.1016/0898-1221(90)90008-8"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282231"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282231/Gaussian_Distributions_Simplifications_and_Simulations"><img alt="Research paper thumbnail of Gaussian Distributions: Simplifications and Simulations" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282231/Gaussian_Distributions_Simplifications_and_Simulations">Gaussian Distributions: Simplifications and Simulations</a></div><div class="wp-workCard_item"><span>Journal of Probability and Statistics</span><span>, 2009</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We present some properties of measures (-Gaussian) that orthogonalize the set of -Hermite polynom...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We present some properties of measures (-Gaussian) that orthogonalize the set of -Hermite polynomials. We also present an algorithm for simulating i.i.d. sequences of random variables having -Gaussian distribution.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282231"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282231"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282231; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282231]").text(description); $(".js-view-count[data-work-id=120282231]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282231; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282231']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282231, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=120282231]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282231,"title":"Gaussian Distributions: Simplifications and Simulations","translated_title":"","metadata":{"abstract":"We present some properties of measures (-Gaussian) that orthogonalize the set of -Hermite polynomials. 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We also present an algorithm for simulating i.i.d. sequences of random variables having -Gaussian distribution.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":16460,"name":"Statistical Physics","url":"https://www.academia.edu/Documents/in/Statistical_Physics"},{"id":342314,"name":"Gaussian","url":"https://www.academia.edu/Documents/in/Gaussian"},{"id":1523776,"name":"Probability statistics","url":"https://www.academia.edu/Documents/in/Probability_statistics"}],"urls":[{"id":42488742,"url":"https://doi.org/10.1155/2009/752430"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282230"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282230/On_the_structure_and_probabilistic_interpretation_of_Askey_Wilson_densities_and_polynomials_with_complex_parameters"><img alt="Research paper thumbnail of On the structure and probabilistic interpretation of Askey–Wilson densities and polynomials with complex parameters" class="work-thumbnail" src="https://attachments.academia-assets.com/115484129/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282230/On_the_structure_and_probabilistic_interpretation_of_Askey_Wilson_densities_and_polynomials_with_complex_parameters">On the structure and probabilistic interpretation of Askey–Wilson densities and polynomials with complex parameters</a></div><div class="wp-workCard_item"><span>Journal of Functional Analysis</span><span>, Aug 1, 2011</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We give equivalent forms of the Askey-Wilson polynomials expressing them with the help of the Al-...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We give equivalent forms of the Askey-Wilson polynomials expressing them with the help of the Al-Salam-Chihara polynomials. After restricting parameters of the Askey-Wilson polynomials to complex conjugate pairs we expand the Askey-Wilson weight function in the series similar to the Poisson-Mehler expansion formula and give its probabilistic interpretation. In particular this result can be used to calculate explicit forms of 'q-Hermite' moments of the Askey-Wilson density, hence enabling calculation of all moments of the Askey-Wilson density. On the way (by setting certain parameter q to 0) we get some formulae useful in the rapidly developing so called 'free probability'.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="76a7b5cbaa2825cb9cdb166e9e01727c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484129,"asset_id":120282230,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484129/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282230"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282230"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282230; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282230]").text(description); $(".js-view-count[data-work-id=120282230]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282230; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282230']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282230, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "76a7b5cbaa2825cb9cdb166e9e01727c" } } $('.js-work-strip[data-work-id=120282230]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282230,"title":"On the structure and probabilistic interpretation of Askey–Wilson densities and polynomials with complex parameters","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"We give equivalent forms of the Askey-Wilson polynomials expressing them with the help of the Al-Salam-Chihara polynomials. 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After restricting parameters of the Askey-Wilson polynomials to complex conjugate pairs we expand the Askey-Wilson weight function in the series similar to the Poisson-Mehler expansion formula and give its probabilistic interpretation. In particular this result can be used to calculate explicit forms of 'q-Hermite' moments of the Askey-Wilson density, hence enabling calculation of all moments of the Askey-Wilson density. On the way (by setting certain parameter q to 0) we get some formulae useful in the rapidly developing so called 'free probability'.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":115484129,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/115484129/thumbnails/1.jpg","file_name":"1011.pdf","download_url":"https://www.academia.edu/attachments/115484129/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_structure_and_probabilistic_inter.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/115484129/1011-libre.pdf?1717097623=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_structure_and_probabilistic_inter.pdf\u0026Expires=1734522074\u0026Signature=GXQy9mEptNlDygBqTlXOV-VD9Uy2IWN~BagKXC3TMFDc4q76tB6KjFSQ9g3zRBiJ-Y7VIhh9h77VdrRhfR0UxaJyObowTCkE7G~4zl-FTHmGfDaT-QxdIzOwGWskZ0FuVZVO1EY60CjblYBCx8248h8rOuUhxNqx9kQTpHaTYxFsoj2UVqZkkdBGkQyc4EtsZhPHfKK0~j1af~sm2Br9xkjgCKPianZqHghSeOmWMIMSCYxA1MmD4otmpOatTRY852B6AEQi0sATXXQF~1gDJmoH-3H95LbEwGPEmsGMIrmu3Vga43qzLunZShqzzaBtxa3GuNr2QfizdzZA57mbjw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":371,"name":"Functional Analysis","url":"https://www.academia.edu/Documents/in/Functional_Analysis"},{"id":12026,"name":"Orthogonal polynomials","url":"https://www.academia.edu/Documents/in/Orthogonal_polynomials"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":346242,"name":"Hermite Polynomials","url":"https://www.academia.edu/Documents/in/Hermite_Polynomials"},{"id":434088,"name":"Free Probability","url":"https://www.academia.edu/Documents/in/Free_Probability"},{"id":2534646,"name":"Chebyshev polynomials","url":"https://www.academia.edu/Documents/in/Chebyshev_polynomials"}],"urls":[{"id":42488741,"url":"https://doi.org/10.1016/j.jfa.2011.04.002"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="120282229"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/120282229/Askey_Wilson_Integral_and_its_Generalizations"><img alt="Research paper thumbnail of Askey--Wilson Integral and its Generalizations" class="work-thumbnail" src="https://attachments.academia-assets.com/115484107/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/120282229/Askey_Wilson_Integral_and_its_Generalizations">Askey--Wilson Integral and its Generalizations</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 20, 2011</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We expand the Askey-Wilson (AW) density in a series of products of continuous q−Hermite polynomia...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We expand the Askey-Wilson (AW) density in a series of products of continuous q−Hermite polynomials times the density that makes these polynomials orthogonal. As a by-product we obtain the value of the AW integral as well as the values of integrals of q−Hermite polynomial times the AW density (q−Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. As a result we obtain system of recurrences that if solved successfully gives a sequence of generalized AW densities with more and more parameters.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="276c20e8b29a856362d73ab51ec8cd1c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":115484107,"asset_id":120282229,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/115484107/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="120282229"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="120282229"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 120282229; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=120282229]").text(description); $(".js-view-count[data-work-id=120282229]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 120282229; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='120282229']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 120282229, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "276c20e8b29a856362d73ab51ec8cd1c" } } $('.js-work-strip[data-work-id=120282229]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":120282229,"title":"Askey--Wilson Integral and its Generalizations","translated_title":"","metadata":{"publisher":"Cornell University","ai_title_tag":"Generalizations of Askey-Wilson Integral and Densities","grobid_abstract":"We expand the Askey-Wilson (AW) density in a series of products of continuous q−Hermite polynomials times the density that makes these polynomials orthogonal. 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As a by-product we obtain the value of the AW integral as well as the values of integrals of q−Hermite polynomial times the AW density (q−Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div><div class="profile--tab_content_container js-tab-pane tab-pane" data-section-id="12952706" id="drafts"><div class="js-work-strip profile--work_container" data-work-id="108670937"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/108670937/ON_POSITIVITY_OF_THE_TWO_PARAMETER_BIVARIATE_KERNEL_BUILT_OF_q_ULTRASPHERICAL_POLYNOMIALS_AND_OTHER_LANCASTER_TYPE_EXPANSIONS_OF_BIVARIATE_DISTRIBUTION"><img alt="Research paper thumbnail of ON POSITIVITY OF THE TWO-PARAMETER BIVARIATE KERNEL BUILT OF q−ULTRASPHERICAL POLYNOMIALS AND OTHER LANCASTER TYPE EXPANSIONS OF BIVARIATE DISTRIBUTION" class="work-thumbnail" src="https://attachments.academia-assets.com/106993132/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/108670937/ON_POSITIVITY_OF_THE_TWO_PARAMETER_BIVARIATE_KERNEL_BUILT_OF_q_ULTRASPHERICAL_POLYNOMIALS_AND_OTHER_LANCASTER_TYPE_EXPANSIONS_OF_BIVARIATE_DISTRIBUTION">ON POSITIVITY OF THE TWO-PARAMETER BIVARIATE KERNEL BUILT OF q−ULTRASPHERICAL POLYNOMIALS AND OTHER LANCASTER TYPE EXPANSIONS OF BIVARIATE DISTRIBUTION</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Our most important result concerns the positivity of certain kernels built of the so-called q−ult...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Our most important result concerns the positivity of certain kernels built of the so-called q−ultraspherical polynomials. Since this result appears at first sight as primarily important for those who are working in orthogonal polynomials, q−series theory and the so-called quantum polynomials, it might have a limited number of interested researchers. That is why, we put our result into a broader context. We recall the theory of Hilbert-Schmidt operators, Lancaster expansions and their applications in Mathematical statistics, or bivariate distributions absolutely continuous with respect to the product of their marginal distributions leading to the generation of Markov process with polynomial conditional moments (the main representative of such processes is a famous Wiener process).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="d0c39ca98a8e7aa45ae1a561aab8c402" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":106993132,"asset_id":108670937,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/106993132/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="108670937"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="108670937"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 108670937; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=108670937]").text(description); $(".js-view-count[data-work-id=108670937]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 108670937; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='108670937']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 108670937, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "d0c39ca98a8e7aa45ae1a561aab8c402" } } $('.js-work-strip[data-work-id=108670937]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":108670937,"title":"ON POSITIVITY OF THE TWO-PARAMETER BIVARIATE KERNEL BUILT OF q−ULTRASPHERICAL POLYNOMIALS AND OTHER LANCASTER TYPE EXPANSIONS OF BIVARIATE DISTRIBUTION","translated_title":"","metadata":{"abstract":"Our most important result concerns the positivity of certain kernels built of the so-called q−ultraspherical polynomials. Since this result appears at first sight as primarily important for those who are working in orthogonal polynomials, q−series theory and the so-called quantum polynomials, it might have a limited number of interested researchers. That is why, we put our result into a broader context. We recall the theory of Hilbert-Schmidt operators, Lancaster expansions and their applications in Mathematical statistics, or bivariate distributions absolutely continuous with respect to the product of their marginal distributions leading to the generation of Markov process with polynomial conditional moments (the main representative of such processes is a famous Wiener process).","ai_title_tag":"Positivity of Bivariate Kernels from q-Ultraspherical Polynomials","publication_date":{"day":null,"month":null,"year":2023,"errors":{}}},"translated_abstract":"Our most important result concerns the positivity of certain kernels built of the so-called q−ultraspherical polynomials. Since this result appears at first sight as primarily important for those who are working in orthogonal polynomials, q−series theory and the so-called quantum polynomials, it might have a limited number of interested researchers. That is why, we put our result into a broader context. 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Since this result appears at first sight as primarily important for those who are working in orthogonal polynomials, q−series theory and the so-called quantum polynomials, it might have a limited number of interested researchers. That is why, we put our result into a broader context. 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We give connection coefficients between them as well as the so-called linearization formulae and other useful important finite and infinite expansions and identities. An important part of the paper is the presentation of probabilistic models where most of these families of polynomials appear. These results were scattered within the literature in recent 15 years. We put them together to enable an overall outlook on these families and understand their crucial rôle in the attempts to generalize Gaussian distributions and find their bounded support generalizations. The paper is based on 65 items in the predominantly recent literature.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="82a5a75e729e3d4e5ff207edfb42a270" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":74883230,"asset_id":61998514,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/74883230/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="61998514"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="61998514"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 61998514; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=61998514]").text(description); $(".js-view-count[data-work-id=61998514]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 61998514; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='61998514']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 61998514, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "82a5a75e729e3d4e5ff207edfb42a270" } } $('.js-work-strip[data-work-id=61998514]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":61998514,"title":"ON THE FAMILIES OF POLYNOMIALS FORMING A PART OF THE ASKEY-WILSON SCHEME AND THEIR PROBABILISTIC APPLICATIONS","translated_title":"","metadata":{"abstract":"We review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey-Wilson scheme of polynomials. 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We give connection coefficients between them as well as the so-called linearization formulae and other useful important finite and infinite expansions and identities. An important part of the paper is the presentation of probabilistic models where most of these families of polynomials appear. These results were scattered within the literature in recent 15 years. We put them together to enable an overall outlook on these families and understand their crucial rôle in the attempts to generalize Gaussian distributions and find their bounded support generalizations. The paper is based on 65 items in the predominantly recent literature.","internal_url":"https://www.academia.edu/61998514/ON_THE_FAMILIES_OF_POLYNOMIALS_FORMING_A_PART_OF_THE_ASKEY_WILSON_SCHEME_AND_THEIR_PROBABILISTIC_APPLICATIONS","translated_internal_url":"","created_at":"2021-11-19T03:37:10.013-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"draft","co_author_tags":[],"downloadable_attachments":[{"id":74883230,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/74883230/thumbnails/1.jpg","file_name":"askey_wilson.pdf","download_url":"https://www.academia.edu/attachments/74883230/download_file?st=MTczNDUxODQ3NCw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"ON_THE_FAMILIES_OF_POLYNOMIALS_FORMING_A.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/74883230/askey_wilson-libre.pdf?1637323897=\u0026response-content-disposition=attachment%3B+filename%3DON_THE_FAMILIES_OF_POLYNOMIALS_FORMING_A.pdf\u0026Expires=1734522074\u0026Signature=O98BAn15gjMNSHQGyG0xfbAiQKpeRNoPJY-HayCRJwrAMG-RTQzj6HCPo6Ml3IDmW6IMZRj4Xw0wFpeZScw7-VP4N-wY5PjUW2XcZBKsiOuspxqfMMP79gxsfiw5yP35gWTeTpyXt-~6eGV65IIO~cG7Q9RAkW0je9mN-ctVjr33GzzGTEVcpffzRro647Hi537TMn6Qbo-kl3liO9KMY4G79DKhoNNCGy9dwEC30dkNzomsCOC4sPsnsOcprXm5QfsJY7SQWmWqdn75u31HmiEGP8o2T3y0Bq92lylG2K-DukJCcJ~Qj3J86Ivlw0xvidqedlRiRcXhed-5PY6gyg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"ON_THE_FAMILIES_OF_POLYNOMIALS_FORMING_A_PART_OF_THE_ASKEY_WILSON_SCHEME_AND_THEIR_PROBABILISTIC_APPLICATIONS","translated_slug":"","page_count":46,"language":"en","content_type":"Work","summary":"We review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey-Wilson scheme of polynomials. 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We present many examples and applications, in analysis and probability. In particular, we give necessary and sufficient conditions for a Lancaster-type of expansion n≥0 cnαn(x)β n (y) with two sets of orthogonal polynomials {αn} and {β n } to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set C(α, β) of the sequences {cn} , for which the above-mentioned series converge to a nonnegative function and give conditions for the membership to it. Further, we show that the class of bivariate distributions for which a Lancaster type expansion can be found, is the same as the class of distributions having all conditional moments in the form of polynomials in the conditioning random variable.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="233d2b2dc73a7d3588e0dea0f5be3513" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":74883056,"asset_id":61998281,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/74883056/download_file?st=MTczNDUxODQ3NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="61998281"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="61998281"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 61998281; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=61998281]").text(description); $(".js-view-count[data-work-id=61998281]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 61998281; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='61998281']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 61998281, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "233d2b2dc73a7d3588e0dea0f5be3513" } } $('.js-work-strip[data-work-id=61998281]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":61998281,"title":"On positivity of orthogonal series and its applications in probability","translated_title":"","metadata":{"abstract":"We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares to a nonnegative function. We present many examples and applications, in analysis and probability. In particular, we give necessary and sufficient conditions for a Lancaster-type of expansion n≥0 cnαn(x)β n (y) with two sets of orthogonal polynomials {αn} and {β n } to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set C(α, β) of the sequences {cn} , for which the above-mentioned series converge to a nonnegative function and give conditions for the membership to it. Further, we show that the class of bivariate distributions for which a Lancaster type expansion can be found, is the same as the class of distributions having all conditional moments in the form of polynomials in the conditioning random variable."},"translated_abstract":"We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares to a nonnegative function. We present many examples and applications, in analysis and probability. In particular, we give necessary and sufficient conditions for a Lancaster-type of expansion n≥0 cnαn(x)β n (y) with two sets of orthogonal polynomials {αn} and {β n } to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set C(α, β) of the sequences {cn} , for which the above-mentioned series converge to a nonnegative function and give conditions for the membership to it. Further, we show that the class of bivariate distributions for which a Lancaster type expansion can be found, is the same as the class of distributions having all conditional moments in the form of polynomials in the conditioning random variable.","internal_url":"https://www.academia.edu/61998281/On_positivity_of_orthogonal_series_and_its_applications_in_probability","translated_internal_url":"","created_at":"2021-11-19T03:34:04.560-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"draft","co_author_tags":[],"downloadable_attachments":[{"id":74883056,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/74883056/thumbnails/1.jpg","file_name":"positivity.pdf","download_url":"https://www.academia.edu/attachments/74883056/download_file?st=MTczNDUxODQ3NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_positivity_of_orthogonal_series_and_i.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/74883056/positivity-libre.pdf?1637323903=\u0026response-content-disposition=attachment%3B+filename%3DOn_positivity_of_orthogonal_series_and_i.pdf\u0026Expires=1734522075\u0026Signature=B5rfFW96HbwQTi2PiCpVX-kgLrqecuhe~~5RAttulu7mqvwxZBLcuFFhnLyBYqIktONwoWC19rv6VuMJtrltIC0PcL0qPJyqhGuH2dofJKKUMTkzbYf-ZPlbdUSf2HhP43Udj7wMtcKIOs2w4MROoJ5gl9sVwVQYK98PAZ2cCZodM6HIiqRlXdDqfbeMcOnLId9qhh80eHCZZdqJvR1aUXkC4BZsFSy3YtaYhjiilxk1zgUkZWpJZogl8mGcahmtYJeM5wsDxasjXy2gI8CPXEkpt~Lm97neQh45YbxYZxdGxR0k2KyUqKBUH7nKXwKAGy0EHz75ruP6-uJz0AXZMQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_positivity_of_orthogonal_series_and_its_applications_in_probability","translated_slug":"","page_count":17,"language":"en","content_type":"Work","summary":"We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares to a nonnegative function. We present many examples and applications, in analysis and probability. In particular, we give necessary and sufficient conditions for a Lancaster-type of expansion n≥0 cnαn(x)β n (y) with two sets of orthogonal polynomials {αn} and {β n } to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set C(α, β) of the sequences {cn} , for which the above-mentioned series converge to a nonnegative function and give conditions for the membership to it. Further, we show that the class of bivariate distributions for which a Lancaster type expansion can be found, is the same as the class of distributions having all conditional moments in the form of polynomials in the conditioning random variable.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":74883056,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/74883056/thumbnails/1.jpg","file_name":"positivity.pdf","download_url":"https://www.academia.edu/attachments/74883056/download_file?st=MTczNDUxODQ3NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_positivity_of_orthogonal_series_and_i.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/74883056/positivity-libre.pdf?1637323903=\u0026response-content-disposition=attachment%3B+filename%3DOn_positivity_of_orthogonal_series_and_i.pdf\u0026Expires=1734522075\u0026Signature=B5rfFW96HbwQTi2PiCpVX-kgLrqecuhe~~5RAttulu7mqvwxZBLcuFFhnLyBYqIktONwoWC19rv6VuMJtrltIC0PcL0qPJyqhGuH2dofJKKUMTkzbYf-ZPlbdUSf2HhP43Udj7wMtcKIOs2w4MROoJ5gl9sVwVQYK98PAZ2cCZodM6HIiqRlXdDqfbeMcOnLId9qhh80eHCZZdqJvR1aUXkC4BZsFSy3YtaYhjiilxk1zgUkZWpJZogl8mGcahmtYJeM5wsDxasjXy2gI8CPXEkpt~Lm97neQh45YbxYZxdGxR0k2KyUqKBUH7nKXwKAGy0EHz75ruP6-uJz0AXZMQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":12026,"name":"Orthogonal polynomials","url":"https://www.academia.edu/Documents/in/Orthogonal_polynomials"},{"id":178344,"name":"Positivity","url":"https://www.academia.edu/Documents/in/Positivity"},{"id":3128247,"name":"Infinite series for mathematical constants","url":"https://www.academia.edu/Documents/in/Infinite_series_for_mathematical_constants"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="61998000"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/61998000/YET_ANOTHER_WAY_OF_CALCULATING_MOMENTS_OF_THE_KESTENS_DISTRIBUTION_AND_ITS_CONSEQUENCES_FOR_CATALAN_NUMBERS_AND_CATALAN_TRIANGLES"><img alt="Research paper thumbnail of YET ANOTHER WAY OF CALCULATING MOMENTS OF THE KESTEN'S DISTRIBUTION AND ITS CONSEQUENCES FOR CATALAN NUMBERS AND CATALAN TRIANGLES" class="work-thumbnail" src="https://attachments.academia-assets.com/74882910/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/61998000/YET_ANOTHER_WAY_OF_CALCULATING_MOMENTS_OF_THE_KESTENS_DISTRIBUTION_AND_ITS_CONSEQUENCES_FOR_CATALAN_NUMBERS_AND_CATALAN_TRIANGLES">YET ANOTHER WAY OF CALCULATING MOMENTS OF THE KESTEN'S DISTRIBUTION AND ITS CONSEQUENCES FOR CATALAN NUMBERS AND CATALAN TRIANGLES</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We calculate moments of the so-called Kesten distribution by means of the expansion of the denomi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles binomial coefficients.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c7e8e522bc354f4398fd1e28f38f66b4" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":74882910,"asset_id":61998000,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/74882910/download_file?st=MTczNDUxODQ3NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="61998000"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="61998000"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 61998000; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=61998000]").text(description); $(".js-view-count[data-work-id=61998000]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 61998000; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='61998000']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 61998000, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c7e8e522bc354f4398fd1e28f38f66b4" } } $('.js-work-strip[data-work-id=61998000]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":61998000,"title":"YET ANOTHER WAY OF CALCULATING MOMENTS OF THE KESTEN'S DISTRIBUTION AND ITS CONSEQUENCES FOR CATALAN NUMBERS AND CATALAN TRIANGLES","translated_title":"","metadata":{"abstract":"We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. 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By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles binomial coefficients.","internal_url":"https://www.academia.edu/61998000/YET_ANOTHER_WAY_OF_CALCULATING_MOMENTS_OF_THE_KESTENS_DISTRIBUTION_AND_ITS_CONSEQUENCES_FOR_CATALAN_NUMBERS_AND_CATALAN_TRIANGLES","translated_internal_url":"","created_at":"2021-11-19T03:28:31.639-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":55567278,"coauthors_can_edit":true,"document_type":"draft","co_author_tags":[],"downloadable_attachments":[{"id":74882910,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/74882910/thumbnails/1.jpg","file_name":"Kesten_i_catalany.pdf","download_url":"https://www.academia.edu/attachments/74882910/download_file?st=MTczNDUxODQ3NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"YET_ANOTHER_WAY_OF_CALCULATING_MOMENTS_O.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/74882910/Kesten_i_catalany-libre.pdf?1637323909=\u0026response-content-disposition=attachment%3B+filename%3DYET_ANOTHER_WAY_OF_CALCULATING_MOMENTS_O.pdf\u0026Expires=1734522075\u0026Signature=YKsk4NTFctmz0YEjWbV2i70ekkZROJqv5mrLNEpetQnYmA~Xtxk66x2o2m05ARqpZ5pgSgEodmfBVX7gycUGDFqSamXPlkNCqF4KfsGxHwSn~nIS4AQKaMrqHIs4fqFebAIjbd0toIvi5ReyMZYlQgtJDg42iiGR0RPJofulqf6badwjJ2u1oCDjp5~HmYjNYHdB5isivQ1cjZTVFm5SX4leoJlao05nf8hz8c7ICVN-lnYRiSJ1jFG~5kkdoevzG8wDLVpa864D4lbFoFvn65HBR9i1j24oILvdokl-vRx6COfSCOa1PY-uCzjgWvd9TY03105bs1yGNFsuyBMCtQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"YET_ANOTHER_WAY_OF_CALCULATING_MOMENTS_OF_THE_KESTENS_DISTRIBUTION_AND_ITS_CONSEQUENCES_FOR_CATALAN_NUMBERS_AND_CATALAN_TRIANGLES","translated_slug":"","page_count":9,"language":"en","content_type":"Work","summary":"We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles binomial coefficients.","owner":{"id":55567278,"first_name":"Paweł","middle_initials":"J","last_name":"Szabłowski","page_name":"PawełSzabłowski","domain_name":"racjonalista","created_at":"2016-10-24T21:57:43.743-07:00","display_name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"},"attachments":[{"id":74882910,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/74882910/thumbnails/1.jpg","file_name":"Kesten_i_catalany.pdf","download_url":"https://www.academia.edu/attachments/74882910/download_file?st=MTczNDUxODQ3NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"YET_ANOTHER_WAY_OF_CALCULATING_MOMENTS_O.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/74882910/Kesten_i_catalany-libre.pdf?1637323909=\u0026response-content-disposition=attachment%3B+filename%3DYET_ANOTHER_WAY_OF_CALCULATING_MOMENTS_O.pdf\u0026Expires=1734522075\u0026Signature=YKsk4NTFctmz0YEjWbV2i70ekkZROJqv5mrLNEpetQnYmA~Xtxk66x2o2m05ARqpZ5pgSgEodmfBVX7gycUGDFqSamXPlkNCqF4KfsGxHwSn~nIS4AQKaMrqHIs4fqFebAIjbd0toIvi5ReyMZYlQgtJDg42iiGR0RPJofulqf6badwjJ2u1oCDjp5~HmYjNYHdB5isivQ1cjZTVFm5SX4leoJlao05nf8hz8c7ICVN-lnYRiSJ1jFG~5kkdoevzG8wDLVpa864D4lbFoFvn65HBR9i1j24oILvdokl-vRx6COfSCOa1PY-uCzjgWvd9TY03105bs1yGNFsuyBMCtQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":344,"name":"Probability Theory","url":"https://www.academia.edu/Documents/in/Probability_Theory"},{"id":309422,"name":"Fibonacci numbers","url":"https://www.academia.edu/Documents/in/Fibonacci_numbers"},{"id":317832,"name":"Lucas numbers","url":"https://www.academia.edu/Documents/in/Lucas_numbers"},{"id":317833,"name":"Catalan numbers","url":"https://www.academia.edu/Documents/in/Catalan_numbers"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="61947033"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/61947033/Extracting_information_from_random_data_Applications_of_laws_of_large_numbers_in_technical_sciences_and_statistics"><img alt="Research paper thumbnail of Extracting information from random data. Applications of laws of large numbers in technical sciences and statistics" class="work-thumbnail" src="https://attachments.academia-assets.com/74848040/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/61947033/Extracting_information_from_random_data_Applications_of_laws_of_large_numbers_in_technical_sciences_and_statistics">Extracting information from random data. Applications of laws of large numbers in technical sciences and statistics</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We formulate conditions for convergence of Laws of Large Numbers and show its links with parts of...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We formulate conditions for convergence of Laws of Large Numbers and show its links with parts of mathematical analysis such as summation theory, the convergence of orthogonal series. Based on this we present applications of the Law of Large Numbers such as Stochastic Approximation, Density and Regression Estimation, Identi…cation.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="e1f3874ab2db8673d36a7de5781ff24a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":74848040,"asset_id":61947033,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/74848040/download_file?st=MTczNDUxODQ3NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="61947033"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="61947033"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 61947033; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=61947033]").text(description); $(".js-view-count[data-work-id=61947033]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 61947033; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='61947033']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 61947033, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "e1f3874ab2db8673d36a7de5781ff24a" } } $('.js-work-strip[data-work-id=61947033]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":61947033,"title":"Extracting information from random data. 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Applications of laws of large numbers in technical sciences and statistics" class="work-thumbnail" src="https://attachments.academia-assets.com/74847098/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/61945102/Extracting_information_from_random_data_Applications_of_laws_of_large_numbers_in_technical_sciences_and_statistics">Extracting information from random data. Applications of laws of large numbers in technical sciences and statistics</a></div><div class="wp-workCard_item"><span>Preprint</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We formulate conditions for convergence of Laws of Large Numbers and show its links with parts of...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We formulate conditions for convergence of Laws of Large Numbers and show its links with parts of mathematical analysis such as summation theory, the convergence of orthogonal series. Based on this we present applications of the Law of Large Numbers such as Stochastic Approximation, Density and Regression Estimation, Identi…cation.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9812846c35bf0af85c41a99d96e5ed27" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":74847098,"asset_id":61945102,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/74847098/download_file?st=MTczNDUxODQ3NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="61945102"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="61945102"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 61945102; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=61945102]").text(description); $(".js-view-count[data-work-id=61945102]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 61945102; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='61945102']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 61945102, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9812846c35bf0af85c41a99d96e5ed27" } } $('.js-work-strip[data-work-id=61945102]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":61945102,"title":"Extracting information from random data. 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