Document Zbl 1508.26006

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M.</a></div> <h2 class="title"> <strong>Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions.</strong> <i>(English)</i> <a class="label nowrap" href="/1508.26006">Zbl 1508.26006</a> </h2> <div class="source"> <a href="/serials/2600" title="Journal Profile">Commun. Nonlinear Sci. Numer. Simul.</a> <a href="/?q=in%3A454308" title="Articles in this Issue">67, 517-527 (2019)</a>; corrigendum ibid. 82, Article ID 104963, 1 p. (2020). </div> <div class="abstract">Summary: We consider an integral transform introduced by <span class="zbmathjax-textit">T. R. Prabhakar</span> [Yokohama Math. J. 19, 7–15 (1971; <a href="/0221.45003">Zbl 0221.45003</a>)], involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann-Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A7263902">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A7263902">Cited in <strong>70</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A26A33" title="MSC2020">26A33</a> </td> <td class="space"> Fractional derivatives and integrals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A33E12" title="MSC2020">33E12</a> </td> <td class="space"> Mittag-Leffler functions and generalizations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A44A10" title="MSC2020">44A10</a> </td> <td class="space"> Laplace transform </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+calculus">fractional calculus</a>; <a href="/?q=ut%3AMittag-Leffler+functions">Mittag-Leffler functions</a>; <a href="/?q=ut%3APrabhakar+operators">Prabhakar operators</a>; <a href="/?q=ut%3Aconvergent+series">convergent series</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0221.45003">Zbl 0221.45003</a></div> <div class="software"> <h3>Software:</h3><a href="/software/20917">DFOC</a>; <a href="/software/20916">sysdfod</a>; <a href="/software/20907">ML</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1508.26006" data-ciurl="/ci/07263902" data-biburl="/bibtex/07263902.bib" data-amsurl="/amsrefs/07263902.bib" data-xmlurl="/xml/07263902.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07263902.pdf" title="Zbl 1508.26006 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.cnsns.2018.07.035" aria-label="DOI for “Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions”" title="10.1016/j.cnsns.2018.07.035">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1807.10101"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Abdeljawad, T.; Baleanu, D., Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J Nonlinear Sci Appl, 10, 3, 1098-1107 (2017) · <a href="/1412.47086" class="nowrap">Zbl 1412.47086</a></td> </tr><tr> <td>[2]</td> <td class="space">Alkahtani, B. 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