CINXE.COM

Appell sequence - Wikipedia

<!doctype html> <html class="client-nojs skin-theme-clientpref-day mf-expand-sections-clientpref-0 mf-font-size-clientpref-small mw-mf-amc-clientpref-0" lang="en" dir="ltr"> <head> <base href="https://en.m.wikipedia.org/wiki/Appell_sequence"> <meta charset="UTF-8"> <title>Appell sequence - Wikipedia</title> <script>(function(){var className="client-js skin-theme-clientpref-day mf-expand-sections-clientpref-0 mf-font-size-clientpref-small mw-mf-amc-clientpref-0";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"3ab3527e-246f-42b0-8193-85498f32559e","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Appell_sequence","wgTitle":"Appell sequence","wgCurRevisionId":1228267574,"wgRevisionId":1228267574,"wgArticleId":1649947, "wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Appell_sequence","wgRelevantArticleId":1649947,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFMode":"stable","wgMFAmc":false,"wgMFAmcOutreachActive":false,"wgMFAmcOutreachUserEligible":false,"wgMFLazyLoadImages":true,"wgMFEditNoticesFeatureConflict":false,"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgMFIsSupportedEditRequest":true,"wgMFScriptPath":"", "wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":7000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgSectionTranslationMissingLanguages":[{"lang":"ace","autonym":"Acèh","dir":"ltr"},{"lang":"ady","autonym":"адыгабзэ","dir":"ltr"},{"lang":"alt","autonym":"алтай тил","dir":"ltr"},{"lang":"am","autonym":"አማርኛ","dir":"ltr"},{"lang":"ami","autonym":"Pangcah","dir":"ltr"},{"lang":"an","autonym":"aragonés","dir":"ltr"},{"lang":"ang","autonym":"Ænglisc","dir":"ltr"},{"lang":"ann","autonym":"Obolo","dir":"ltr"},{"lang":"anp","autonym":"अंगिका","dir":"ltr"},{"lang":"ar","autonym":"العربية","dir":"rtl"},{"lang":"ary","autonym":"الدارجة","dir":"rtl"},{"lang":"arz","autonym":"مصرى","dir":"rtl"},{"lang":"as","autonym":"অসমীয়া","dir":"ltr"},{"lang":"ast","autonym":"asturianu","dir":"ltr"},{"lang":"av","autonym":"авар","dir":"ltr"},{"lang":"avk","autonym":"Kotava","dir":"ltr"},{"lang":"awa" ,"autonym":"अवधी","dir":"ltr"},{"lang":"ay","autonym":"Aymar aru","dir":"ltr"},{"lang":"az","autonym":"azərbaycanca","dir":"ltr"},{"lang":"azb","autonym":"تۆرکجه","dir":"rtl"},{"lang":"ba","autonym":"башҡортса","dir":"ltr"},{"lang":"ban","autonym":"Basa Bali","dir":"ltr"},{"lang":"bar","autonym":"Boarisch","dir":"ltr"},{"lang":"bbc","autonym":"Batak Toba","dir":"ltr"},{"lang":"bcl","autonym":"Bikol Central","dir":"ltr"},{"lang":"bdr","autonym":"Bajau Sama","dir":"ltr"},{"lang":"be","autonym":"беларуская","dir":"ltr"},{"lang":"bew","autonym":"Betawi","dir":"ltr"},{"lang":"bg","autonym":"български","dir":"ltr"},{"lang":"bho","autonym":"भोजपुरी","dir":"ltr"},{"lang":"bi","autonym":"Bislama","dir":"ltr"},{"lang":"bjn","autonym":"Banjar","dir":"ltr"},{"lang":"blk","autonym":"ပအိုဝ်ႏဘာႏသာႏ","dir":"ltr"},{"lang":"bm","autonym":"bamanankan","dir":"ltr"},{"lang":"bn","autonym":"বাংলা","dir":"ltr"},{ "lang":"bo","autonym":"བོད་ཡིག","dir":"ltr"},{"lang":"bpy","autonym":"বিষ্ণুপ্রিয়া মণিপুরী","dir":"ltr"},{"lang":"br","autonym":"brezhoneg","dir":"ltr"},{"lang":"bs","autonym":"bosanski","dir":"ltr"},{"lang":"btm","autonym":"Batak Mandailing","dir":"ltr"},{"lang":"bug","autonym":"Basa Ugi","dir":"ltr"},{"lang":"ca","autonym":"català","dir":"ltr"},{"lang":"cdo","autonym":"閩東語 / Mìng-dĕ̤ng-ngṳ̄","dir":"ltr"},{"lang":"ce","autonym":"нохчийн","dir":"ltr"},{"lang":"ceb","autonym":"Cebuano","dir":"ltr"},{"lang":"ch","autonym":"Chamoru","dir":"ltr"},{"lang":"chr","autonym":"ᏣᎳᎩ","dir":"ltr"},{"lang":"ckb","autonym":"کوردی","dir":"rtl"},{"lang":"co","autonym":"corsu","dir":"ltr"},{"lang":"cr","autonym":"Nēhiyawēwin / ᓀᐦᐃᔭᐍᐏᐣ","dir":"ltr"},{"lang":"crh","autonym":"qırımtatarca","dir":"ltr"},{"lang":"cu","autonym":"словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ","dir":"ltr"},{"lang": "cy","autonym":"Cymraeg","dir":"ltr"},{"lang":"da","autonym":"dansk","dir":"ltr"},{"lang":"dag","autonym":"dagbanli","dir":"ltr"},{"lang":"de","autonym":"Deutsch","dir":"ltr"},{"lang":"dga","autonym":"Dagaare","dir":"ltr"},{"lang":"din","autonym":"Thuɔŋjäŋ","dir":"ltr"},{"lang":"diq","autonym":"Zazaki","dir":"ltr"},{"lang":"dsb","autonym":"dolnoserbski","dir":"ltr"},{"lang":"dtp","autonym":"Kadazandusun","dir":"ltr"},{"lang":"dv","autonym":"ދިވެހިބަސް","dir":"rtl"},{"lang":"dz","autonym":"ཇོང་ཁ","dir":"ltr"},{"lang":"ee","autonym":"eʋegbe","dir":"ltr"},{"lang":"el","autonym":"Ελληνικά","dir":"ltr"},{"lang":"eml","autonym":"emiliàn e rumagnòl","dir":"ltr"},{"lang":"eo","autonym":"Esperanto","dir":"ltr"},{"lang":"et","autonym":"eesti","dir":"ltr"},{"lang":"eu","autonym":"euskara","dir":"ltr"},{"lang":"fa","autonym":"فارسی","dir":"rtl"},{"lang":"fat","autonym":"mfantse","dir":"ltr"},{"lang":"ff","autonym":"Fulfulde","dir":"ltr"},{"lang":"fi", "autonym":"suomi","dir":"ltr"},{"lang":"fj","autonym":"Na Vosa Vakaviti","dir":"ltr"},{"lang":"fo","autonym":"føroyskt","dir":"ltr"},{"lang":"fon","autonym":"fɔ̀ngbè","dir":"ltr"},{"lang":"fr","autonym":"français","dir":"ltr"},{"lang":"frp","autonym":"arpetan","dir":"ltr"},{"lang":"frr","autonym":"Nordfriisk","dir":"ltr"},{"lang":"fur","autonym":"furlan","dir":"ltr"},{"lang":"fy","autonym":"Frysk","dir":"ltr"},{"lang":"gag","autonym":"Gagauz","dir":"ltr"},{"lang":"gan","autonym":"贛語","dir":"ltr"},{"lang":"gcr","autonym":"kriyòl gwiyannen","dir":"ltr"},{"lang":"gl","autonym":"galego","dir":"ltr"},{"lang":"glk","autonym":"گیلکی","dir":"rtl"},{"lang":"gn","autonym":"Avañe'ẽ","dir":"ltr"},{"lang":"gom","autonym":"गोंयची कोंकणी / Gõychi Konknni","dir":"ltr"},{"lang":"gor","autonym":"Bahasa Hulontalo","dir":"ltr"},{"lang":"gpe","autonym":"Ghanaian Pidgin","dir":"ltr"},{"lang":"gu","autonym":"ગુજરાતી","dir":"ltr"},{"lang":"guc", "autonym":"wayuunaiki","dir":"ltr"},{"lang":"gur","autonym":"farefare","dir":"ltr"},{"lang":"guw","autonym":"gungbe","dir":"ltr"},{"lang":"gv","autonym":"Gaelg","dir":"ltr"},{"lang":"ha","autonym":"Hausa","dir":"ltr"},{"lang":"hak","autonym":"客家語 / Hak-kâ-ngî","dir":"ltr"},{"lang":"haw","autonym":"Hawaiʻi","dir":"ltr"},{"lang":"he","autonym":"עברית","dir":"rtl"},{"lang":"hi","autonym":"हिन्दी","dir":"ltr"},{"lang":"hif","autonym":"Fiji Hindi","dir":"ltr"},{"lang":"hr","autonym":"hrvatski","dir":"ltr"},{"lang":"hsb","autonym":"hornjoserbsce","dir":"ltr"},{"lang":"ht","autonym":"Kreyòl ayisyen","dir":"ltr"},{"lang":"hu","autonym":"magyar","dir":"ltr"},{"lang":"hy","autonym":"հայերեն","dir":"ltr"},{"lang":"hyw","autonym":"Արեւմտահայերէն","dir":"ltr"},{"lang":"ia","autonym":"interlingua","dir":"ltr"},{"lang":"iba","autonym":"Jaku Iban","dir":"ltr"},{"lang":"ie","autonym":"Interlingue","dir":"ltr"},{"lang":"ig","autonym":"Igbo","dir":"ltr"},{ "lang":"igl","autonym":"Igala","dir":"ltr"},{"lang":"ilo","autonym":"Ilokano","dir":"ltr"},{"lang":"io","autonym":"Ido","dir":"ltr"},{"lang":"is","autonym":"íslenska","dir":"ltr"},{"lang":"it","autonym":"italiano","dir":"ltr"},{"lang":"iu","autonym":"ᐃᓄᒃᑎᑐᑦ / inuktitut","dir":"ltr"},{"lang":"ja","autonym":"日本語","dir":"ltr"},{"lang":"jam","autonym":"Patois","dir":"ltr"},{"lang":"jv","autonym":"Jawa","dir":"ltr"},{"lang":"ka","autonym":"ქართული","dir":"ltr"},{"lang":"kaa","autonym":"Qaraqalpaqsha","dir":"ltr"},{"lang":"kab","autonym":"Taqbaylit","dir":"ltr"},{"lang":"kbd","autonym":"адыгэбзэ","dir":"ltr"},{"lang":"kbp","autonym":"Kabɩyɛ","dir":"ltr"},{"lang":"kcg","autonym":"Tyap","dir":"ltr"},{"lang":"kg","autonym":"Kongo","dir":"ltr"},{"lang":"kge","autonym":"Kumoring","dir":"ltr"},{"lang":"ki","autonym":"Gĩkũyũ","dir":"ltr"},{"lang":"kk","autonym":"қазақша","dir":"ltr"},{"lang":"kl","autonym":"kalaallisut","dir":"ltr"},{"lang":"km", "autonym":"ភាសាខ្មែរ","dir":"ltr"},{"lang":"kn","autonym":"ಕನ್ನಡ","dir":"ltr"},{"lang":"ko","autonym":"한국어","dir":"ltr"},{"lang":"koi","autonym":"перем коми","dir":"ltr"},{"lang":"krc","autonym":"къарачай-малкъар","dir":"ltr"},{"lang":"ks","autonym":"कॉशुर / کٲشُر","dir":"rtl"},{"lang":"ku","autonym":"kurdî","dir":"ltr"},{"lang":"kus","autonym":"Kʋsaal","dir":"ltr"},{"lang":"kv","autonym":"коми","dir":"ltr"},{"lang":"kw","autonym":"kernowek","dir":"ltr"},{"lang":"ky","autonym":"кыргызча","dir":"ltr"},{"lang":"lad","autonym":"Ladino","dir":"ltr"},{"lang":"lb","autonym":"Lëtzebuergesch","dir":"ltr"},{"lang":"lez","autonym":"лезги","dir":"ltr"},{"lang":"lg","autonym":"Luganda","dir":"ltr"},{"lang":"li","autonym":"Limburgs","dir":"ltr"},{"lang":"lij","autonym":"Ligure","dir":"ltr"},{"lang":"lld","autonym":"Ladin","dir":"ltr"},{"lang":"lmo","autonym":"lombard","dir":"ltr"},{"lang":"ln","autonym": "lingála","dir":"ltr"},{"lang":"lo","autonym":"ລາວ","dir":"ltr"},{"lang":"lt","autonym":"lietuvių","dir":"ltr"},{"lang":"ltg","autonym":"latgaļu","dir":"ltr"},{"lang":"lv","autonym":"latviešu","dir":"ltr"},{"lang":"mad","autonym":"Madhurâ","dir":"ltr"},{"lang":"mai","autonym":"मैथिली","dir":"ltr"},{"lang":"map-bms","autonym":"Basa Banyumasan","dir":"ltr"},{"lang":"mdf","autonym":"мокшень","dir":"ltr"},{"lang":"mg","autonym":"Malagasy","dir":"ltr"},{"lang":"mhr","autonym":"олык марий","dir":"ltr"},{"lang":"mi","autonym":"Māori","dir":"ltr"},{"lang":"min","autonym":"Minangkabau","dir":"ltr"},{"lang":"mk","autonym":"македонски","dir":"ltr"},{"lang":"ml","autonym":"മലയാളം","dir":"ltr"},{"lang":"mn","autonym":"монгол","dir":"ltr"},{"lang":"mni","autonym":"ꯃꯤꯇꯩ ꯂꯣꯟ","dir":"ltr"},{"lang":"mnw","autonym":"ဘာသာမန်","dir":"ltr"},{"lang":"mos","autonym":"moore","dir":"ltr"},{"lang":"mr","autonym": "मराठी","dir":"ltr"},{"lang":"mrj","autonym":"кырык мары","dir":"ltr"},{"lang":"ms","autonym":"Bahasa Melayu","dir":"ltr"},{"lang":"mt","autonym":"Malti","dir":"ltr"},{"lang":"mwl","autonym":"Mirandés","dir":"ltr"},{"lang":"my","autonym":"မြန်မာဘာသာ","dir":"ltr"},{"lang":"myv","autonym":"эрзянь","dir":"ltr"},{"lang":"mzn","autonym":"مازِرونی","dir":"rtl"},{"lang":"nah","autonym":"Nāhuatl","dir":"ltr"},{"lang":"nan","autonym":"閩南語 / Bân-lâm-gú","dir":"ltr"},{"lang":"nap","autonym":"Napulitano","dir":"ltr"},{"lang":"nb","autonym":"norsk bokmål","dir":"ltr"},{"lang":"nds","autonym":"Plattdüütsch","dir":"ltr"},{"lang":"nds-nl","autonym":"Nedersaksies","dir":"ltr"},{"lang":"ne","autonym":"नेपाली","dir":"ltr"},{"lang":"new","autonym":"नेपाल भाषा","dir":"ltr"},{"lang":"nia","autonym":"Li Niha","dir":"ltr"},{"lang":"nl","autonym":"Nederlands","dir":"ltr"},{"lang":"nn","autonym":"norsk nynorsk", "dir":"ltr"},{"lang":"nqo","autonym":"ߒߞߏ","dir":"rtl"},{"lang":"nr","autonym":"isiNdebele seSewula","dir":"ltr"},{"lang":"nso","autonym":"Sesotho sa Leboa","dir":"ltr"},{"lang":"ny","autonym":"Chi-Chewa","dir":"ltr"},{"lang":"oc","autonym":"occitan","dir":"ltr"},{"lang":"om","autonym":"Oromoo","dir":"ltr"},{"lang":"or","autonym":"ଓଡ଼ିଆ","dir":"ltr"},{"lang":"os","autonym":"ирон","dir":"ltr"},{"lang":"pa","autonym":"ਪੰਜਾਬੀ","dir":"ltr"},{"lang":"pag","autonym":"Pangasinan","dir":"ltr"},{"lang":"pam","autonym":"Kapampangan","dir":"ltr"},{"lang":"pap","autonym":"Papiamentu","dir":"ltr"},{"lang":"pcd","autonym":"Picard","dir":"ltr"},{"lang":"pcm","autonym":"Naijá","dir":"ltr"},{"lang":"pdc","autonym":"Deitsch","dir":"ltr"},{"lang":"pl","autonym":"polski","dir":"ltr"},{"lang":"pms","autonym":"Piemontèis","dir":"ltr"},{"lang":"pnb","autonym":"پنجابی","dir":"rtl"},{"lang":"ps","autonym":"پښتو","dir":"rtl"},{"lang":"pt","autonym":"português","dir": "ltr"},{"lang":"pwn","autonym":"pinayuanan","dir":"ltr"},{"lang":"qu","autonym":"Runa Simi","dir":"ltr"},{"lang":"rm","autonym":"rumantsch","dir":"ltr"},{"lang":"rn","autonym":"ikirundi","dir":"ltr"},{"lang":"ro","autonym":"română","dir":"ltr"},{"lang":"rsk","autonym":"руски","dir":"ltr"},{"lang":"rue","autonym":"русиньскый","dir":"ltr"},{"lang":"rup","autonym":"armãneashti","dir":"ltr"},{"lang":"rw","autonym":"Ikinyarwanda","dir":"ltr"},{"lang":"sa","autonym":"संस्कृतम्","dir":"ltr"},{"lang":"sah","autonym":"саха тыла","dir":"ltr"},{"lang":"sat","autonym":"ᱥᱟᱱᱛᱟᱲᱤ","dir":"ltr"},{"lang":"sc","autonym":"sardu","dir":"ltr"},{"lang":"scn","autonym":"sicilianu","dir":"ltr"},{"lang":"sco","autonym":"Scots","dir":"ltr"},{"lang":"sd","autonym":"سنڌي","dir":"rtl"},{"lang":"se","autonym":"davvisámegiella","dir":"ltr"},{"lang":"sg","autonym":"Sängö","dir":"ltr"},{"lang":"sgs","autonym":"žemaitėška","dir":"ltr"},{"lang":"sh", "autonym":"srpskohrvatski / српскохрватски","dir":"ltr"},{"lang":"shi","autonym":"Taclḥit","dir":"ltr"},{"lang":"shn","autonym":"ၽႃႇသႃႇတႆး ","dir":"ltr"},{"lang":"si","autonym":"සිංහල","dir":"ltr"},{"lang":"skr","autonym":"سرائیکی","dir":"rtl"},{"lang":"sl","autonym":"slovenščina","dir":"ltr"},{"lang":"sm","autonym":"Gagana Samoa","dir":"ltr"},{"lang":"smn","autonym":"anarâškielâ","dir":"ltr"},{"lang":"sn","autonym":"chiShona","dir":"ltr"},{"lang":"so","autonym":"Soomaaliga","dir":"ltr"},{"lang":"sq","autonym":"shqip","dir":"ltr"},{"lang":"srn","autonym":"Sranantongo","dir":"ltr"},{"lang":"ss","autonym":"SiSwati","dir":"ltr"},{"lang":"st","autonym":"Sesotho","dir":"ltr"},{"lang":"stq","autonym":"Seeltersk","dir":"ltr"},{"lang":"su","autonym":"Sunda","dir":"ltr"},{"lang":"sv","autonym":"svenska","dir":"ltr"},{"lang":"sw","autonym":"Kiswahili","dir":"ltr"},{"lang":"szl","autonym":"ślůnski","dir":"ltr"},{"lang":"ta","autonym": "தமிழ்","dir":"ltr"},{"lang":"tay","autonym":"Tayal","dir":"ltr"},{"lang":"tcy","autonym":"ತುಳು","dir":"ltr"},{"lang":"tdd","autonym":"ᥖᥭᥰ ᥖᥬᥲ ᥑᥨᥒᥰ","dir":"ltr"},{"lang":"te","autonym":"తెలుగు","dir":"ltr"},{"lang":"tet","autonym":"tetun","dir":"ltr"},{"lang":"tg","autonym":"тоҷикӣ","dir":"ltr"},{"lang":"th","autonym":"ไทย","dir":"ltr"},{"lang":"ti","autonym":"ትግርኛ","dir":"ltr"},{"lang":"tig","autonym":"ትግሬ","dir":"ltr"},{"lang":"tk","autonym":"Türkmençe","dir":"ltr"},{"lang":"tl","autonym":"Tagalog","dir":"ltr"},{"lang":"tly","autonym":"tolışi","dir":"ltr"},{"lang":"tn","autonym":"Setswana","dir":"ltr"},{"lang":"to","autonym":"lea faka-Tonga","dir":"ltr"},{"lang":"tpi","autonym":"Tok Pisin","dir":"ltr"},{"lang":"tr","autonym":"Türkçe","dir":"ltr"},{"lang":"trv","autonym":"Seediq","dir":"ltr"},{"lang":"ts","autonym":"Xitsonga","dir":"ltr"},{"lang":"tt","autonym":"татарча / tatarça","dir":"ltr"},{ "lang":"tum","autonym":"chiTumbuka","dir":"ltr"},{"lang":"tw","autonym":"Twi","dir":"ltr"},{"lang":"ty","autonym":"reo tahiti","dir":"ltr"},{"lang":"tyv","autonym":"тыва дыл","dir":"ltr"},{"lang":"udm","autonym":"удмурт","dir":"ltr"},{"lang":"ur","autonym":"اردو","dir":"rtl"},{"lang":"uz","autonym":"oʻzbekcha / ўзбекча","dir":"ltr"},{"lang":"ve","autonym":"Tshivenda","dir":"ltr"},{"lang":"vec","autonym":"vèneto","dir":"ltr"},{"lang":"vep","autonym":"vepsän kel’","dir":"ltr"},{"lang":"vi","autonym":"Tiếng Việt","dir":"ltr"},{"lang":"vls","autonym":"West-Vlams","dir":"ltr"},{"lang":"vo","autonym":"Volapük","dir":"ltr"},{"lang":"vro","autonym":"võro","dir":"ltr"},{"lang":"wa","autonym":"walon","dir":"ltr"},{"lang":"war","autonym":"Winaray","dir":"ltr"},{"lang":"wo","autonym":"Wolof","dir":"ltr"},{"lang":"wuu","autonym":"吴语","dir":"ltr"},{"lang":"xal","autonym":"хальмг","dir":"ltr"},{"lang":"xh","autonym":"isiXhosa","dir":"ltr"},{"lang":"xmf", "autonym":"მარგალური","dir":"ltr"},{"lang":"yi","autonym":"ייִדיש","dir":"rtl"},{"lang":"yo","autonym":"Yorùbá","dir":"ltr"},{"lang":"yue","autonym":"粵語","dir":"ltr"},{"lang":"za","autonym":"Vahcuengh","dir":"ltr"},{"lang":"zgh","autonym":"ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ","dir":"ltr"},{"lang":"zu","autonym":"isiZulu","dir":"ltr"}],"wgSectionTranslationTargetLanguages":["ace","ady","alt","am","ami","an","ang","ann","anp","ar","ary","arz","as","ast","av","avk","awa","ay","az","azb","ba","ban","bar","bbc","bcl","bdr","be","bew","bg","bho","bi","bjn","blk","bm","bn","bo","bpy","br","bs","btm","bug","ca","cdo","ce","ceb","ch","chr","ckb","co","cr","crh","cs","cu","cy","da","dag","de","dga","din","diq","dsb","dtp","dv","dz","ee","el","eml","eo","es","et","eu","fa","fat","ff","fi","fj","fo","fon","fr","frp","frr","fur","fy","gag","gan","gcr","gl","glk","gn","gom","gor","gpe","gu","guc","gur","guw","gv","ha","hak","haw","he","hi","hif","hr","hsb" ,"ht","hu","hy","hyw","ia","iba","ie","ig","igl","ilo","io","is","it","iu","ja","jam","jv","ka","kaa","kab","kbd","kbp","kcg","kg","kge","ki","kk","kl","km","kn","ko","koi","krc","ks","ku","kus","kv","kw","ky","lad","lb","lez","lg","li","lij","lld","lmo","ln","lo","lt","ltg","lv","mad","mai","map-bms","mdf","mg","mhr","mi","min","mk","ml","mn","mni","mnw","mos","mr","mrj","ms","mt","mwl","my","myv","mzn","nah","nan","nap","nb","nds","nds-nl","ne","new","nia","nl","nn","nqo","nr","nso","ny","oc","om","or","os","pa","pag","pam","pap","pcd","pcm","pdc","pl","pms","pnb","ps","pt","pwn","qu","rm","rn","ro","rsk","rue","rup","rw","sa","sah","sat","sc","scn","sco","sd","se","sg","sgs","sh","shi","shn","si","sk","skr","sl","sm","smn","sn","so","sq","sr","srn","ss","st","stq","su","sv","sw","szl","ta","tay","tcy","tdd","te","tet","tg","th","ti","tig","tk","tl","tly","tn","to","tpi","tr","trv","ts","tt","tum","tw","ty","tyv","udm","ur","uz","ve","vec","vep","vi","vls","vo","vro","wa","war","wo", "wuu","xal","xh","xmf","yi","yo","yue","za","zgh","zh","zu"],"isLanguageSearcherCXEntrypointEnabled":true,"mintEntrypointLanguages":["ace","ast","azb","bcl","bjn","bh","crh","ff","fon","ig","is","ki","ks","lmo","min","sat","ss","tn","vec"],"wgWikibaseItemId":"Q1090038","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false,"wgMinervaPermissions":{"watchable":true,"watch":false},"wgMinervaFeatures":{"beta":false,"donate":true,"mobileOptionsLink":true,"categories":false,"pageIssues":true,"talkAtTop":true,"historyInPageActions":false,"overflowSubmenu":false,"tabsOnSpecials":true,"personalMenu":false,"mainMenuExpanded":false,"echo":true,"nightMode":true},"wgMinervaDownloadNamespaces":[0]};RLSTATE={"ext.globalCssJs.user.styles":"ready", "site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","skins.minerva.styles":"ready","skins.minerva.content.styles.images":"ready","mediawiki.hlist":"ready","skins.minerva.codex.styles":"ready","skins.minerva.icons":"ready","skins.minerva.amc.styles":"ready","ext.wikimediamessages.styles":"ready","mobile.init.styles":"ready","ext.relatedArticles.styles":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["site","mediawiki.page.ready","skins.minerva.scripts","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","ext.popups","mobile.init","ext.echo.centralauth","ext.relatedArticles.readMore.bootstrap","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.cx.eventlogging.campaigns","ext.cx.entrypoints.mffrequentlanguages","ext.cx.entrypoints.languagesearcher.init", "mw.externalguidance.init","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&amp;modules=ext.math.styles%7Cext.relatedArticles.styles%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cmediawiki.hlist%7Cmobile.init.styles%7Cskins.minerva.amc.styles%7Cskins.minerva.codex.styles%7Cskins.minerva.content.styles.images%7Cskins.minerva.icons%2Cstyles%7Cwikibase.client.init&amp;only=styles&amp;skin=minerva"> <script async src="/w/load.php?lang=en&amp;modules=startup&amp;only=scripts&amp;raw=1&amp;skin=minerva"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&amp;modules=site.styles&amp;only=styles&amp;skin=minerva"> <meta name="generator" content="MediaWiki 1.44.0-wmf.12"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="theme-color" content="#eaecf0"> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes, minimum-scale=0.25, maximum-scale=5.0"> <meta property="og:title" content="Appell sequence - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="manifest" href="/w/api.php?action=webapp-manifest"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Appell_sequence&amp;action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Appell_sequence"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="dns-prefetch" href="//meta.wikimedia.org"> <link rel="dns-prefetch" href="login.wikimedia.org"> <meta http-equiv="X-Translated-By" content="Google"> <meta http-equiv="X-Translated-To" content="en"> <script type="text/javascript" src="https://www.gstatic.com/_/translate_http/_/js/k=translate_http.tr.en_GB.20k2T9bCXYg.O/am=BgM/d=1/rs=AN8SPfrBTbfinOztiAdztux_N6C7bXZBHg/m=corsproxy" data-sourceurl="https://en.m.wikipedia.org/wiki/Appell_sequence"></script> <link href="https://fonts.googleapis.com/css2?family=Material+Symbols+Outlined:opsz,wght,FILL,GRAD@20..48,100..700,0..1,-50..200" rel="stylesheet"> <script type="text/javascript" src="https://www.gstatic.com/_/translate_http/_/js/k=translate_http.tr.en_GB.20k2T9bCXYg.O/am=BgM/d=1/exm=corsproxy/ed=1/rs=AN8SPfrBTbfinOztiAdztux_N6C7bXZBHg/m=phishing_protection" data-phishing-protection-enabled="false" data-forms-warning-enabled="true" data-source-url="https://en.m.wikipedia.org/wiki/Appell_sequence"></script> <meta name="robots" content="none"> </head> <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Appell_sequence rootpage-Appell_sequence stable issues-group-B skin-minerva action-view skin--responsive mw-mf-amc-disabled mw-mf"> <script type="text/javascript" src="https://www.gstatic.com/_/translate_http/_/js/k=translate_http.tr.en_GB.20k2T9bCXYg.O/am=BgM/d=1/exm=corsproxy,phishing_protection/ed=1/rs=AN8SPfrBTbfinOztiAdztux_N6C7bXZBHg/m=navigationui" data-environment="prod" data-proxy-url="https://en-m-wikipedia-org.translate.goog" data-proxy-full-url="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-source-url="https://en.m.wikipedia.org/wiki/Appell_sequence" data-source-language="auto" data-target-language="en" data-display-language="en-GB" data-detected-source-language="en" data-is-source-untranslated="false" data-source-untranslated-url="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://en.m.wikipedia.org/wiki/Appell_sequence&amp;anno=2" data-client="tr"></script> <div id="mw-mf-viewport"> <div id="mw-mf-page-center"><a class="mw-mf-page-center__mask" href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#"></a> <header class="header-container header-chrome"> <div class="minerva-header"> <nav class="navigation-drawer toggle-list view-border-box"><input type="checkbox" id="main-menu-input" class="toggle-list__checkbox" role="button" aria-haspopup="true" aria-expanded="false" aria-labelledby="mw-mf-main-menu-button"> <label role="button" for="main-menu-input" id="mw-mf-main-menu-button" aria-hidden="true" data-event-name="ui.mainmenu" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet toggle-list__toggle"> <span class="minerva-icon minerva-icon--menu"></span> <span></span> </label> <div id="mw-mf-page-left" class="menu view-border-box"> <ul id="p-navigation" class="toggle-list__list"> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--home" href="https://en-m-wikipedia-org.translate.goog/wiki/Main_Page?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-mw="interface"> <span class="minerva-icon minerva-icon--home"></span> <span class="toggle-list-item__label">Home</span> </a></li> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--random" href="https://en-m-wikipedia-org.translate.goog/wiki/Special:Random?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-mw="interface"> <span class="minerva-icon minerva-icon--die"></span> <span class="toggle-list-item__label">Random</span> </a></li> <li class="toggle-list-item skin-minerva-list-item-jsonly"><a class="toggle-list-item__anchor menu__item--nearby" href="https://en-m-wikipedia-org.translate.goog/wiki/Special:Nearby?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-event-name="menu.nearby" data-mw="interface"> <span class="minerva-icon minerva-icon--mapPin"></span> <span class="toggle-list-item__label">Nearby</span> </a></li> </ul> <ul id="p-personal" class="toggle-list__list"> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--login" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Special:UserLogin&amp;returnto=Appell+sequence&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-event-name="menu.login" data-mw="interface"> <span class="minerva-icon minerva-icon--logIn"></span> <span class="toggle-list-item__label">Log in</span> </a></li> </ul> <ul id="pt-preferences" class="toggle-list__list"> <li class="toggle-list-item skin-minerva-list-item-jsonly"><a class="toggle-list-item__anchor menu__item--settings" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Special:MobileOptions&amp;returnto=Appell+sequence&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-event-name="menu.settings" data-mw="interface"> <span class="minerva-icon minerva-icon--settings"></span> <span class="toggle-list-item__label">Settings</span> </a></li> </ul> <ul id="p-donation" class="toggle-list__list"> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--donate" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://donate.wikimedia.org/?wmf_source%3Ddonate%26wmf_medium%3Dsidebar%26wmf_campaign%3Den.wikipedia.org%26uselang%3Den%26wmf_key%3Dminerva" data-event-name="menu.donate" data-mw="interface"> <span class="minerva-icon minerva-icon--heart"></span> <span class="toggle-list-item__label">Donate</span> </a></li> </ul> <ul class="hlist"> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--about" href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:About?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-mw="interface"> <span class="toggle-list-item__label">About Wikipedia</span> </a></li> <li class="toggle-list-item "><a class="toggle-list-item__anchor menu__item--disclaimers" href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:General_disclaimer?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-mw="interface"> <span class="toggle-list-item__label">Disclaimers</span> </a></li> </ul> </div><label class="main-menu-mask" for="main-menu-input"></label> </nav> <div class="branding-box"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Main_Page?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB"> <span><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"> </span> </a> </div> <form action="/w/index.php" method="get" class="minerva-search-form"> <div class="search-box"><input type="hidden" name="title" value="Special:Search"> <input class="search skin-minerva-search-trigger" id="searchInput" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f"> <span class="search-box-icon-overlay"><span class="minerva-icon minerva-icon--search"></span> </span> </div><button id="searchIcon" class="cdx-button cdx-button--size-large cdx-button--icon-only cdx-button--weight-quiet skin-minerva-search-trigger"> <span class="minerva-icon minerva-icon--search"></span> <span>Search</span> </button> </form> <nav class="minerva-user-navigation" aria-label="User navigation"> </nav> </div> </header> <main id="content" class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Appell sequence</span></h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"><a class="minerva__tab-text" href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" rel="" data-event-name="tabs.subject">Article</a></li> <li class="minerva__tab "><a class="minerva__tab-text" href="https://en-m-wikipedia-org.translate.goog/wiki/Talk:Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" rel="discussion" data-event-name="tabs.talk">Talk</a></li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"><a role="button" href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> <span class="minerva-icon minerva-icon--language"></span> <span>Language</span> </a></li> <li id="page-actions-watch" class="page-actions-menu__list-item"><a role="button" id="ca-watch" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Special:UserLogin&amp;returnto=Appell+sequence&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> <span class="minerva-icon minerva-icon--star"></span> <span>Watch</span> </a></li> <li id="page-actions-edit" class="page-actions-menu__list-item"><a role="button" id="ca-edit" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=edit&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Edit</span> </a></li> </ul> </nav><!-- version 1.0.2 (change every time you update a partial) --> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style> <table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"> <tbody> <tr> <td class="mbox-text"> <div class="mbox-text-span"> This article includes a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:Citing_sources?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Wikipedia:Citing sources">list of references</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:Further_reading?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Wikipedia:Further reading">related reading</a>, or <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:External_links?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:Citing_sources?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:When_to_cite?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">May 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="https://en-m-wikipedia-org.translate.goog/wiki/Help:Maintenance_template_removal?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span> </div></td> </tr> </tbody> </table> <p>In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematics?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Mathematics">mathematics</a>, an <b>Appell sequence</b>, named after <a href="https://en-m-wikipedia-org.translate.goog/wiki/Paul_%C3%89mile_Appell?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Paul Émile Appell">Paul Émile Appell</a>, is any <a href="https://en-m-wikipedia-org.translate.goog/wiki/Polynomial_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Polynomial sequence">polynomial sequence</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p_{n}(x)\}_{n=0,1,2,\ldots }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <msub> <mo fence="false" stretchy="false"> } </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mo> …<!-- … --> </mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{p_{n}(x)\}_{n=0,1,2,\ldots }} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17c0e3b3dd02a5c63ddbfb3e8a7dfa194f64beb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.113ex; height:3.009ex;" alt="{\displaystyle \{p_{n}(x)\}_{n=0,1,2,\ldots }}"></span> satisfying the identity</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mi> x </mi> </mrow> </mfrac> </mrow> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> n </mi> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x),} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/accb3ebecb789a420052392a7ed6b2f30ef02c76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.676ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x),}"></span> </dd> </dl> <p>and in which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{0}(x)} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54df21a553ecfaf1415edfed2c340114894a9f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.452ex; height:2.843ex;" alt="{\displaystyle p_{0}(x)}"></span> is a non-zero constant.</p> <p>Among the most notable Appell sequences besides the trivial example <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x^{n}\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{x^{n}\}} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4176c4eaaf6d824861c1dcd3103f59dbd3f1f14d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.873ex; height:2.843ex;" alt="{\displaystyle \{x^{n}\}}"></span> are the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hermite_polynomials?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Hermite polynomials">Hermite polynomials</a>, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bernoulli_polynomials?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Bernoulli polynomials">Bernoulli polynomials</a>, and the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euler_polynomials?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" class="mw-redirect" title="Euler polynomials">Euler polynomials</a>. Every Appell sequence is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sheffer_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Sheffer sequence">Sheffer sequence</a>, but most Sheffer sequences are not Appell sequences. Appell sequences have a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Probability_theory?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Probability theory">probabilistic</a> interpretation as systems of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Moment_(mathematics)?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Moment (mathematics)">moments</a>.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#Equivalent_characterizations_of_Appell_sequences"><span class="tocnumber">1</span> <span class="toctext">Equivalent characterizations of Appell sequences</span></a></li> <li class="toclevel-1 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#Recursion_formula"><span class="tocnumber">2</span> <span class="toctext">Recursion formula</span></a></li> <li class="toclevel-1 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#Subgroup_of_the_Sheffer_polynomials"><span class="tocnumber">3</span> <span class="toctext">Subgroup of the Sheffer polynomials</span></a></li> <li class="toclevel-1 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#Different_convention"><span class="tocnumber">4</span> <span class="toctext">Different convention</span></a></li> <li class="toclevel-1 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#Hypergeometric_Appell_polynomials"><span class="tocnumber">5</span> <span class="toctext">Hypergeometric Appell polynomials</span></a></li> <li class="toclevel-1 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#See_also"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#External_links"><span class="tocnumber">8</span> <span class="toctext">External links</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Equivalent_characterizations_of_Appell_sequences">Equivalent characterizations of Appell sequences</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=edit&amp;section=1&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Edit section: Equivalent characterizations of Appell sequences" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>The following conditions on polynomial sequences can easily be seen to be equivalent:</p> <ul> <li>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1,2,3,\ldots }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> <mo> , </mo> <mo> …<!-- … --> </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n=1,2,3,\ldots } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a508e267228dcaa193fe3a7df64c94b0f31e143e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\displaystyle n=1,2,3,\ldots }"> </noscript><span class="lazy-image-placeholder" style="width: 13.806ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a508e267228dcaa193fe3a7df64c94b0f31e143e" data-alt="{\displaystyle n=1,2,3,\ldots }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>,</li> </ul> <dl> <dd> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mi> x </mi> </mrow> </mfrac> </mrow> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> n </mi> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b51c7dd7222c906acecfc820ccee54ca75169d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.029ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 21.029ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b51c7dd7222c906acecfc820ccee54ca75169d" data-alt="{\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> </dd> </dl> <dl> <dd> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{0}(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54df21a553ecfaf1415edfed2c340114894a9f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.452ex; height:2.843ex;" alt="{\displaystyle p_{0}(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 5.452ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54df21a553ecfaf1415edfed2c340114894a9f3" data-alt="{\displaystyle p_{0}(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> is a non-zero constant; </dd> </dl> <ul> <li>For some sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{c_{n}\}_{n=0}^{\infty }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false"> } </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\textstyle \{c_{n}\}_{n=0}^{\infty }} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f38153162f3c60d95e186f5b3f29f24ec4709a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.869ex; height:3.009ex;" alt="{\textstyle \{c_{n}\}_{n=0}^{\infty }}"> </noscript><span class="lazy-image-placeholder" style="width: 7.869ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f38153162f3c60d95e186f5b3f29f24ec4709a" data-alt="{\textstyle \{c_{n}\}_{n=0}^{\infty }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> of scalars with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}\neq 0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ≠<!-- ≠ --> </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle c_{0}\neq 0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ad029670a8a84a14f07f1ae7e1c5536a9ca54b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.322ex; height:2.676ex;" alt="{\displaystyle c_{0}\neq 0}"> </noscript><span class="lazy-image-placeholder" style="width: 6.322ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ad029670a8a84a14f07f1ae7e1c5536a9ca54b" data-alt="{\displaystyle c_{0}\neq 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>,</li> </ul> <dl> <dd> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}c_{k}x^{n-k};}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mi> k </mi> </mrow> </msup> <mo> ; </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}c_{k}x^{n-k};} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caece7d70b70fdc16660e1def1640d2142fdfbce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:24.698ex; height:7.009ex;" alt="{\displaystyle p_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}c_{k}x^{n-k};}"> </noscript><span class="lazy-image-placeholder" style="width: 24.698ex;height: 7.009ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caece7d70b70fdc16660e1def1640d2142fdfbce" data-alt="{\displaystyle p_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}c_{k}x^{n-k};}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> </dd> </dl> <ul> <li>For the same sequence of scalars,</li> </ul> <dl> <dd> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <msup> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8898e3f617fb68b1fafd42f8fc70cefd2b6288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:25.665ex; height:7.509ex;" alt="{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},}"> </noscript><span class="lazy-image-placeholder" style="width: 25.665ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8898e3f617fb68b1fafd42f8fc70cefd2b6288" data-alt="{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> </dd> </dl> <dl> <dd> where </dd> </dl> <dl> <dd> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\frac {d}{dx}};}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> D </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mi> x </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle D={\frac {d}{dx}};} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f51b64ed22f039abe1164648903e91c8d52b27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.051ex; height:5.509ex;" alt="{\displaystyle D={\frac {d}{dx}};}"> </noscript><span class="lazy-image-placeholder" style="width: 9.051ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f51b64ed22f039abe1164648903e91c8d52b27" data-alt="{\displaystyle D={\frac {d}{dx}};}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> </dd> </dl> <ul> <li>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0,1,2,\ldots }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mo> …<!-- … --> </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n=0,1,2,\ldots } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a19cb2cfd4f9ebdbc8e5cbb9b92ecb9ace85cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\displaystyle n=0,1,2,\ldots }"> </noscript><span class="lazy-image-placeholder" style="width: 13.806ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a19cb2cfd4f9ebdbc8e5cbb9b92ecb9ace85cab" data-alt="{\displaystyle n=0,1,2,\ldots }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>,</li> </ul> <dl> <dd> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)y^{n-k}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mi> k </mi> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)y^{n-k}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c535f158644d8cda970f081c8e17baff9d6312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:31.827ex; height:7.009ex;" alt="{\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)y^{n-k}.}"> </noscript><span class="lazy-image-placeholder" style="width: 31.827ex;height: 7.009ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c535f158644d8cda970f081c8e17baff9d6312" data-alt="{\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)y^{n-k}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Recursion_formula">Recursion formula</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=edit&amp;section=2&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Edit section: Recursion formula" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Suppose</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{c_{k} \over k!}D^{k}\right)x^{n}=Sx^{n},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <msup> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> = </mo> <mi> S </mi> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{c_{k} \over k!}D^{k}\right)x^{n}=Sx^{n},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e0314a470a1cd48fe88bb88cac5cdcfbef909b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:32.811ex; height:7.509ex;" alt="{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{c_{k} \over k!}D^{k}\right)x^{n}=Sx^{n},}"> </noscript><span class="lazy-image-placeholder" style="width: 32.811ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e0314a470a1cd48fe88bb88cac5cdcfbef909b" data-alt="{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{c_{k} \over k!}D^{k}\right)x^{n}=Sx^{n},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>where the last equality is taken to define the linear operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> </noscript><span class="lazy-image-placeholder" style="width: 1.499ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" data-alt="{\displaystyle S}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> on the space of polynomials in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>. Let</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=S^{-1}=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)^{-1}=\sum _{k=1}^{\infty }{\frac {a_{k}}{k!}}D^{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> T </mi> <mo> = </mo> <msup> <mi> S </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <msup> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <msup> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T=S^{-1}=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)^{-1}=\sum _{k=1}^{\infty }{\frac {a_{k}}{k!}}D^{k}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/262c0d56228c515195c94ad4423f05237a663796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.397ex; height:8.009ex;" alt="{\displaystyle T=S^{-1}=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)^{-1}=\sum _{k=1}^{\infty }{\frac {a_{k}}{k!}}D^{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 40.397ex;height: 8.009ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/262c0d56228c515195c94ad4423f05237a663796" data-alt="{\displaystyle T=S^{-1}=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)^{-1}=\sum _{k=1}^{\infty }{\frac {a_{k}}{k!}}D^{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>be the inverse operator, the coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{k}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.319ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" data-alt="{\displaystyle a_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> being those of the usual reciprocal of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Formal_power_series?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Formal power series">formal power series</a>, so that</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tp_{n}(x)=x^{n}.\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> T </mi> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> . </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle Tp_{n}(x)=x^{n}.\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e072244fd376dabd069ec7dc0fcd14481b452f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.844ex; height:2.843ex;" alt="{\displaystyle Tp_{n}(x)=x^{n}.\,}"> </noscript><span class="lazy-image-placeholder" style="width: 13.844ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e072244fd376dabd069ec7dc0fcd14481b452f50" data-alt="{\displaystyle Tp_{n}(x)=x^{n}.\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>In the conventions of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Umbral_calculus?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Umbral calculus">umbral calculus</a>, one often treats this formal <a href="https://en-m-wikipedia-org.translate.goog/wiki/Power_series?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Power series">power series</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> T </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle T} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> </noscript><span class="lazy-image-placeholder" style="width: 1.636ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" data-alt="{\displaystyle T}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> as representing the Appell sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.477ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" data-alt="{\displaystyle p_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>. One can define</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log T=\log \left(\sum _{k=0}^{\infty }{\frac {a_{k}}{k!}}D^{k}\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> log </mi> <mo> ⁡<!-- ⁡ --> </mo> <mi> T </mi> <mo> = </mo> <mi> log </mi> <mo> ⁡<!-- ⁡ --> </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <msup> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \log T=\log \left(\sum _{k=0}^{\infty }{\frac {a_{k}}{k!}}D^{k}\right)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67297dbbbdd76416832b2079984f70fe4030fc25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.656ex; height:7.509ex;" alt="{\displaystyle \log T=\log \left(\sum _{k=0}^{\infty }{\frac {a_{k}}{k!}}D^{k}\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 24.656ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67297dbbbdd76416832b2079984f70fe4030fc25" data-alt="{\displaystyle \log T=\log \left(\sum _{k=0}^{\infty }{\frac {a_{k}}{k!}}D^{k}\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>by using the usual power series expansion of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> log </mi> <mo> ⁡<!-- ⁡ --> </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \log(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4157d3b51ac7b147fca145d431d58ec92abc1f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.111ex; height:2.843ex;" alt="{\displaystyle \log(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 6.111ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4157d3b51ac7b147fca145d431d58ec92abc1f70" data-alt="{\displaystyle \log(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> and the usual definition of composition of formal power series. Then we have</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n+1}(x)=(x-(\log T)')p_{n}(x).\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> −<!-- − --> </mo> <mo stretchy="false"> ( </mo> <mi> log </mi> <mo> ⁡<!-- ⁡ --> </mo> <mi> T </mi> <msup> <mo stretchy="false"> ) </mo> <mo> ′ </mo> </msup> <mo stretchy="false"> ) </mo> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> . </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{n+1}(x)=(x-(\log T)')p_{n}(x).\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/222a42ebae9df93d70c4c5bab57c323b3ad4c994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:30.845ex; height:3.009ex;" alt="{\displaystyle p_{n+1}(x)=(x-(\log T)')p_{n}(x).\,}"> </noscript><span class="lazy-image-placeholder" style="width: 30.845ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/222a42ebae9df93d70c4c5bab57c323b3ad4c994" data-alt="{\displaystyle p_{n+1}(x)=(x-(\log T)')p_{n}(x).\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>(This formal differentiation of a power series in the differential operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> D </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle D} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> </noscript><span class="lazy-image-placeholder" style="width: 1.924ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" data-alt="{\displaystyle D}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> is an instance of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pincherle_derivative?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Pincherle derivative">Pincherle differentiation</a>.)</p> <p>In the case of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hermite_polynomials?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Hermite polynomials">Hermite polynomials</a>, this reduces to the conventional recursion formula for that sequence.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Subgroup_of_the_Sheffer_polynomials">Subgroup of the Sheffer polynomials</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=edit&amp;section=3&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Edit section: Subgroup of the Sheffer polynomials" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>The set of all Appell sequences is closed under the operation of umbral composition of polynomial sequences, defined as follows. Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p_{n}(x)\colon n=0,1,2,\ldots \}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> :<!-- : --> </mo> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mo> …<!-- … --> </mo> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{p_{n}(x)\colon n=0,1,2,\ldots \}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ab64df68da6ad9963a13896ff9e1a31e805c74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.691ex; height:2.843ex;" alt="{\displaystyle \{p_{n}(x)\colon n=0,1,2,\ldots \}}"> </noscript><span class="lazy-image-placeholder" style="width: 22.691ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ab64df68da6ad9963a13896ff9e1a31e805c74" data-alt="{\displaystyle \{p_{n}(x)\colon n=0,1,2,\ldots \}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{q_{n}(x)\colon n=0,1,2,\ldots \}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> q </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> :<!-- : --> </mo> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mo> …<!-- … --> </mo> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{q_{n}(x)\colon n=0,1,2,\ldots \}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ead981defe8e70d3ad9c82f1c58f974c0a389f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.559ex; height:2.843ex;" alt="{\displaystyle \{q_{n}(x)\colon n=0,1,2,\ldots \}}"> </noscript><span class="lazy-image-placeholder" style="width: 22.559ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ead981defe8e70d3ad9c82f1c58f974c0a389f37" data-alt="{\displaystyle \{q_{n}(x)\colon n=0,1,2,\ldots \}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> are polynomial sequences, given by</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)=\sum _{k=0}^{n}a_{n,k}x^{k}{\text{ and }}q_{n}(x)=\sum _{k=0}^{n}b_{n,k}x^{k}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> &nbsp;and&nbsp; </mtext> </mrow> <msub> <mi> q </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{n}(x)=\sum _{k=0}^{n}a_{n,k}x^{k}{\text{ and }}q_{n}(x)=\sum _{k=0}^{n}b_{n,k}x^{k}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a19df0f10d7929db5f412ed1cc3e188ee058ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:42.376ex; height:7.009ex;" alt="{\displaystyle p_{n}(x)=\sum _{k=0}^{n}a_{n,k}x^{k}{\text{ and }}q_{n}(x)=\sum _{k=0}^{n}b_{n,k}x^{k}.}"> </noscript><span class="lazy-image-placeholder" style="width: 42.376ex;height: 7.009ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a19df0f10d7929db5f412ed1cc3e188ee058ca5" data-alt="{\displaystyle p_{n}(x)=\sum _{k=0}^{n}a_{n,k}x^{k}{\text{ and }}q_{n}(x)=\sum _{k=0}^{n}b_{n,k}x^{k}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>Then the umbral composition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\circ q}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo> ∘<!-- ∘ --> </mo> <mi> q </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p\circ q} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22db961ff7d40ef4500f7f1eee858f696d404097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.523ex; height:2.009ex;" alt="{\displaystyle p\circ q}"> </noscript><span class="lazy-image-placeholder" style="width: 4.523ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22db961ff7d40ef4500f7f1eee858f696d404097" data-alt="{\displaystyle p\circ q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> is the polynomial sequence whose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>th term is</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p_{n}\circ q)(x)=\sum _{k=0}^{n}a_{n,k}q_{k}(x)=\sum _{0\leq \ell \leq k\leq n}a_{n,k}b_{k,\ell }x^{\ell }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> ∘<!-- ∘ --> </mo> <mi> q </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <msub> <mi> q </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munder> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> <mo> ≤<!-- ≤ --> </mo> <mi> ℓ<!-- ℓ --> </mi> <mo> ≤<!-- ≤ --> </mo> <mi> k </mi> <mo> ≤<!-- ≤ --> </mo> <mi> n </mi> </mrow> </munder> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> , </mo> <mi> ℓ<!-- ℓ --> </mi> </mrow> </msub> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> ℓ<!-- ℓ --> </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (p_{n}\circ q)(x)=\sum _{k=0}^{n}a_{n,k}q_{k}(x)=\sum _{0\leq \ell \leq k\leq n}a_{n,k}b_{k,\ell }x^{\ell }} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78db9a80a8fca26f05711aace97e1159c1f99671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.378ex; height:7.176ex;" alt="{\displaystyle (p_{n}\circ q)(x)=\sum _{k=0}^{n}a_{n,k}q_{k}(x)=\sum _{0\leq \ell \leq k\leq n}a_{n,k}b_{k,\ell }x^{\ell }}"> </noscript><span class="lazy-image-placeholder" style="width: 46.378ex;height: 7.176ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78db9a80a8fca26f05711aace97e1159c1f99671" data-alt="{\displaystyle (p_{n}\circ q)(x)=\sum _{k=0}^{n}a_{n,k}q_{k}(x)=\sum _{0\leq \ell \leq k\leq n}a_{n,k}b_{k,\ell }x^{\ell }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>(the subscript <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> appears in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\displaystyle p_{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.477ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f79dcba35ecde0d43fbb7c914165586166ce8c2" data-alt="{\displaystyle p_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>, since this is the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>th term of that sequence, but not in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> q </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle q} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> </noscript><span class="lazy-image-placeholder" style="width: 1.07ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" data-alt="{\displaystyle q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>, since this refers to the sequence as a whole rather than one of its terms).</p> <p>Under this operation, the set of all Sheffer sequences is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Non-abelian_group?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Non-abelian group">non-abelian group</a>, but the set of all Appell sequences is an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Abelian_group?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Abelian group">abelian</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Subgroup?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Subgroup">subgroup</a>. That it is abelian can be seen by considering the fact that every Appell sequence is of the form</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <msup> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8898e3f617fb68b1fafd42f8fc70cefd2b6288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:25.665ex; height:7.509ex;" alt="{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},}"> </noscript><span class="lazy-image-placeholder" style="width: 25.665ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8898e3f617fb68b1fafd42f8fc70cefd2b6288" data-alt="{\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>and that umbral composition of Appell sequences corresponds to multiplication of these <a href="https://en-m-wikipedia-org.translate.goog/wiki/Formal_power_series?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Formal power series">formal power series</a> in the operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> D </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle D} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> </noscript><span class="lazy-image-placeholder" style="width: 1.924ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" data-alt="{\displaystyle D}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Different_convention">Different convention</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=edit&amp;section=4&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Edit section: Different convention" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"> <table class="box-Format_footnotes plainlinks metadata ambox ambox-style" role="presentation"> <tbody> <tr> <td class="mbox-text"> <div class="mbox-text-span"> This section includes <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:CITE?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB#Types_of_citation" class="mw-redirect" title="Wikipedia:CITE">inline citations</a>, but <b>they are not <a href="https://en-m-wikipedia-org.translate.goog/wiki/Help:Footnotes?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Help:Footnotes">properly formatted</a>.</b><span class="hide-when-compact"> Please <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <span class="plainlinks"><a class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://en.wikipedia.org/w/index.php?title%3DAppell_sequence%26%26action%3Dedit">correcting them</a></span>.</span> <span class="date-container"><i>(<span class="date">May 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="https://en-m-wikipedia-org.translate.goog/wiki/Help:Maintenance_template_removal?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span> </div></td> </tr> </tbody> </table> <p>Another convention followed by some authors (see <i>Chihara</i>) defines this concept in a different way, conflicting with Appell's original definition, by using the identity</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {d \over dx}p_{n}(x)=p_{n-1}(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mi> x </mi> </mrow> </mfrac> </mrow> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {d \over dx}p_{n}(x)=p_{n-1}(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55563f44a25d3eee193c5efcf419ececbdf05bdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.634ex; height:5.509ex;" alt="{\displaystyle {d \over dx}p_{n}(x)=p_{n-1}(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 19.634ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55563f44a25d3eee193c5efcf419ececbdf05bdc" data-alt="{\displaystyle {d \over dx}p_{n}(x)=p_{n-1}(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>instead.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Hypergeometric_Appell_polynomials">Hypergeometric Appell polynomials</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=edit&amp;section=5&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Edit section: Hypergeometric Appell polynomials" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <p>The enormous class of Appell polynomials can be obtained in terms of the generalized hypergeometric function.</p> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (k,-n)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Δ<!-- Δ --> </mi> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> , </mo> <mo> −<!-- − --> </mo> <mi> n </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Delta (k,-n)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99c6edd056e203c9956656e10f24001f89318bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \Delta (k,-n)}"> </noscript><span class="lazy-image-placeholder" style="width: 9.193ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99c6edd056e203c9956656e10f24001f89318bd2" data-alt="{\displaystyle \Delta (k,-n)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> denote the array of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> </noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> ratios</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {n}{k}},-{\frac {n-1}{k}},\ldots ,-{\frac {n-k+1}{k}},\quad n\in {\mathbb {N} }_{0},k\in \mathbb {N} .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> n </mi> <mi> k </mi> </mfrac> </mrow> <mo> , </mo> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> , </mo> <mo> …<!-- … --> </mo> <mo> , </mo> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <mo> −<!-- − --> </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mi> n </mi> <mo> ∈<!-- ∈ --> </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> N </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mi> k </mi> <mo> ∈<!-- ∈ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> N </mi> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -{\frac {n}{k}},-{\frac {n-1}{k}},\ldots ,-{\frac {n-k+1}{k}},\quad n\in {\mathbb {N} }_{0},k\in \mathbb {N} .} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acde91d380139bcdafa80579800e68f15063119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:48.121ex; height:5.509ex;" alt="{\displaystyle -{\frac {n}{k}},-{\frac {n-1}{k}},\ldots ,-{\frac {n-k+1}{k}},\quad n\in {\mathbb {N} }_{0},k\in \mathbb {N} .}"> </noscript><span class="lazy-image-placeholder" style="width: 48.121ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acde91d380139bcdafa80579800e68f15063119" data-alt="{\displaystyle -{\frac {n}{k}},-{\frac {n-1}{k}},\ldots ,-{\frac {n-k+1}{k}},\quad n\in {\mathbb {N} }_{0},k\in \mathbb {N} .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> </dd> </dl> <p>Consider the polynomial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n,p,q}^{(k)}(a,b;m,x)=x^{n}{}_{k+p}F_{q}\left({a_{1}},{a_{2}},\ldots ,{a_{p}},\Delta (k,-n);{b_{1}},{b_{2}},\ldots ,{b_{q}};{\frac {m}{x^{k}}}\right),\quad n,m\in \mathbb {N} _{0},k\in \mathbb {N} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> , </mo> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> ; </mo> <mi> m </mi> <mo> , </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> </msub> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> q </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mo> …<!-- … --> </mo> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> </mrow> <mo> , </mo> <mi mathvariant="normal"> Δ<!-- Δ --> </mi> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> , </mo> <mo> −<!-- − --> </mo> <mi> n </mi> <mo stretchy="false"> ) </mo> <mo> ; </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mo> …<!-- … --> </mo> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> q </mi> </mrow> </msub> </mrow> <mo> ; </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> m </mi> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mi> n </mi> <mo> , </mo> <mi> m </mi> <mo> ∈<!-- ∈ --> </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mi> k </mi> <mo> ∈<!-- ∈ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> N </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{n,p,q}^{(k)}(a,b;m,x)=x^{n}{}_{k+p}F_{q}\left({a_{1}},{a_{2}},\ldots ,{a_{p}},\Delta (k,-n);{b_{1}},{b_{2}},\ldots ,{b_{q}};{\frac {m}{x^{k}}}\right),\quad n,m\in \mathbb {N} _{0},k\in \mathbb {N} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f986c2f711298691bd2cdcf1fcac719ba3ae0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:92.57ex; height:6.176ex;" alt="{\displaystyle A_{n,p,q}^{(k)}(a,b;m,x)=x^{n}{}_{k+p}F_{q}\left({a_{1}},{a_{2}},\ldots ,{a_{p}},\Delta (k,-n);{b_{1}},{b_{2}},\ldots ,{b_{q}};{\frac {m}{x^{k}}}\right),\quad n,m\in \mathbb {N} _{0},k\in \mathbb {N} }"> </noscript><span class="lazy-image-placeholder" style="width: 92.57ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f986c2f711298691bd2cdcf1fcac719ba3ae0cc" data-alt="{\displaystyle A_{n,p,q}^{(k)}(a,b;m,x)=x^{n}{}_{k+p}F_{q}\left({a_{1}},{a_{2}},\ldots ,{a_{p}},\Delta (k,-n);{b_{1}},{b_{2}},\ldots ,{b_{q}};{\frac {m}{x^{k}}}\right),\quad n,m\in \mathbb {N} _{0},k\in \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span></p> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{k+p}F_{q}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> </msub> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> q </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {}_{k+p}F_{q}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c31d0537046918169c8120dec27134edd72fcdd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.677ex; height:2.843ex;" alt="{\displaystyle {}_{k+p}F_{q}}"> </noscript><span class="lazy-image-placeholder" style="width: 5.677ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c31d0537046918169c8120dec27134edd72fcdd0" data-alt="{\displaystyle {}_{k+p}F_{q}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> is the generalized hypergeometric function.</p> <p><b>Theorem.</b> <i>The polynomial family <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A_{n,p,q}^{(k)}(a,b;m,x)\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msubsup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> , </mo> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> ; </mo> <mi> m </mi> <mo> , </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{A_{n,p,q}^{(k)}(a,b;m,x)\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b093400bc9490c802cc8e7942ca443e6a14992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.293ex; height:3.676ex;" alt="{\displaystyle \{A_{n,p,q}^{(k)}(a,b;m,x)\}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.293ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b093400bc9490c802cc8e7942ca443e6a14992" data-alt="{\displaystyle \{A_{n,p,q}^{(k)}(a,b;m,x)\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> is the Appell sequence for any natural parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,p,q,m,k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> p </mi> <mo> , </mo> <mi> q </mi> <mo> , </mo> <mi> m </mi> <mo> , </mo> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a,b,p,q,m,k} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636ed266c4839ff84a62e868e824a0e657b88a72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.888ex; height:2.509ex;" alt="{\displaystyle a,b,p,q,m,k}"> </noscript><span class="lazy-image-placeholder" style="width: 12.888ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636ed266c4839ff84a62e868e824a0e657b88a72" data-alt="{\displaystyle a,b,p,q,m,k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>.</i></p> <p>For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0,q=0,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p=0,q=0,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9fbb8225e771d7f28338ab2a1c7a2f7ab48815e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.531ex; height:2.509ex;" alt="{\displaystyle p=0,q=0,}"> </noscript><span class="lazy-image-placeholder" style="width: 12.531ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9fbb8225e771d7f28338ab2a1c7a2f7ab48815e" data-alt="{\displaystyle p=0,q=0,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=m,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> <mo> = </mo> <mi> m </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k=m,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/599ce0706083e949789635ee8eebf996b15d3b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.997ex; height:2.509ex;" alt="{\displaystyle k=m,}"> </noscript><span class="lazy-image-placeholder" style="width: 6.997ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/599ce0706083e949789635ee8eebf996b15d3b91" data-alt="{\displaystyle k=m,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=(-1)^{k}h{k^{k}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <mo> = </mo> <mo stretchy="false"> ( </mo> <mo> −<!-- − --> </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mi> h </mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m=(-1)^{k}h{k^{k}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70173a2e5ab841ed8c52d9349b8e921cf83241ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.646ex; height:3.176ex;" alt="{\displaystyle m=(-1)^{k}h{k^{k}}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.646ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70173a2e5ab841ed8c52d9349b8e921cf83241ab" data-alt="{\displaystyle m=(-1)^{k}h{k^{k}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> then the polynomials <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n,p,q}^{(k)}(m,x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> , </mo> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo stretchy="false"> ( </mo> <mi> m </mi> <mo> , </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{n,p,q}^{(k)}(m,x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f02a5f8125ffde699ebfb9c9fc705b0f5f1048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.673ex; height:3.676ex;" alt="{\displaystyle A_{n,p,q}^{(k)}(m,x)}"> </noscript><span class="lazy-image-placeholder" style="width: 11.673ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f02a5f8125ffde699ebfb9c9fc705b0f5f1048" data-alt="{\displaystyle A_{n,p,q}^{(k)}(m,x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> become the Gould-Hopper polynomials <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{n}^{m}(x,h)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi> g </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msubsup> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> h </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle g_{n}^{m}(x,h)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec1ca8b10e33ddf755c5d98d4a0a6a91197d529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.305ex; height:2.843ex;" alt="{\displaystyle g_{n}^{m}(x,h)}"> </noscript><span class="lazy-image-placeholder" style="width: 8.305ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec1ca8b10e33ddf755c5d98d4a0a6a91197d529" data-alt="{\displaystyle g_{n}^{m}(x,h)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> and if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0,q=0,m=-2,k=2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mi> m </mi> <mo> = </mo> <mo> −<!-- − --> </mo> <mn> 2 </mn> <mo> , </mo> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p=0,q=0,m=-2,k=2} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aea27827a34d828d785aa8840194b26966622c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:27.534ex; height:2.509ex;" alt="{\displaystyle p=0,q=0,m=-2,k=2}"> </noscript><span class="lazy-image-placeholder" style="width: 27.534ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aea27827a34d828d785aa8840194b26966622c6" data-alt="{\displaystyle p=0,q=0,m=-2,k=2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span> they become the Hermite polynomials <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle H_{n}(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505cd70a83ef6433715abc22c4d2ed86058c2738" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.289ex; height:2.843ex;" alt="{\displaystyle H_{n}(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 6.289ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505cd70a83ef6433715abc22c4d2ed86058c2738" data-alt="{\displaystyle H_{n}(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">&nbsp;</span></span>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=edit&amp;section=6&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Sheffer_sequence?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Sheffer sequence">Sheffer sequence</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Umbral_calculus?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Umbral calculus">Umbral calculus</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Generalized_Appell_polynomials?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Generalized Appell polynomials">Generalized Appell polynomials</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Wick_product?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Wick product">Wick product</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=edit&amp;section=7&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <ul> <li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAppell1880" class="citation journal cs1">Appell, Paul (1880). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=http://www.numdam.org/item?id%3DASENS_1880_2_9__119_0">"Sur une classe de polynômes"</a>. <i>Annales Scientifiques de l'École Normale Supérieure</i>. 2e Série. <b>9</b>: <span class="nowrap">119–</span>144. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://doi.org/10.24033%252Fasens.186">10.24033/asens.186</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Annales+Scientifiques+de+l%27%C3%89cole+Normale+Sup%C3%A9rieure&amp;rft.atitle=Sur+une+classe+de+polyn%C3%B4mes&amp;rft.volume=9&amp;rft.pages=%3Cspan+class%3D%22nowrap%22%3E119-%3C%2Fspan%3E144&amp;rft.date=1880&amp;rft_id=info%3Adoi%2F10.24033%2Fasens.186&amp;rft.aulast=Appell&amp;rft.aufirst=Paul&amp;rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DASENS_1880_2_9__119_0&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAppell+sequence" class="Z3988"></span></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRomanRota1978" class="citation journal cs1">Roman, Steven; Rota, Gian-Carlo (1978). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://doi.org/10.1016%252F0001-8708%252878%252990087-7">"The Umbral Calculus"</a>. <i><a href="https://en-m-wikipedia-org.translate.goog/wiki/Advances_in_Mathematics?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Advances in Mathematics">Advances in Mathematics</a></i>. <b>27</b> (2): <span class="nowrap">95–</span>188. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://doi.org/10.1016%252F0001-8708%252878%252990087-7">10.1016/0001-8708(78)90087-7</a></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Advances+in+Mathematics&amp;rft.atitle=The+Umbral+Calculus&amp;rft.volume=27&amp;rft.issue=2&amp;rft.pages=%3Cspan+class%3D%22nowrap%22%3E95-%3C%2Fspan%3E188&amp;rft.date=1978&amp;rft_id=info%3Adoi%2F10.1016%2F0001-8708%2878%2990087-7&amp;rft.aulast=Roman&amp;rft.aufirst=Steven&amp;rft.au=Rota%2C+Gian-Carlo&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0001-8708%252878%252990087-7&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAppell+sequence" class="Z3988"></span>.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotaKahanerOdlyzko1973" class="citation journal cs1">Rota, Gian-Carlo; Kahaner, D.; Odlyzko, Andrew (1973). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://doi.org/10.1016%252F0022-247X%252873%252990172-8">"Finite Operator Calculus"</a>. <i>Journal of Mathematical Analysis and Applications</i>. <b>42</b> (3): <span class="nowrap">685–</span>760. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://doi.org/10.1016%252F0022-247X%252873%252990172-8">10.1016/0022-247X(73)90172-8</a></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Journal+of+Mathematical+Analysis+and+Applications&amp;rft.atitle=Finite+Operator+Calculus&amp;rft.volume=42&amp;rft.issue=3&amp;rft.pages=%3Cspan+class%3D%22nowrap%22%3E685-%3C%2Fspan%3E760&amp;rft.date=1973&amp;rft_id=info%3Adoi%2F10.1016%2F0022-247X%2873%2990172-8&amp;rft.aulast=Rota&amp;rft.aufirst=Gian-Carlo&amp;rft.au=Kahaner%2C+D.&amp;rft.au=Odlyzko%2C+Andrew&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0022-247X%252873%252990172-8&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAppell+sequence" class="Z3988"></span> Reprinted in the book with the same title, Academic Press, New York, 1975.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteven_Roman" class="citation book cs1">Steven Roman. <i>The Umbral Calculus</i>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dover_Publications?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Dover Publications">Dover Publications</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Umbral+Calculus&amp;rft.pub=Dover+Publications&amp;rft.au=Steven+Roman&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAppell+sequence" class="Z3988"></span></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTheodore_Seio_Chihara1978" class="citation book cs1">Theodore Seio Chihara (1978). <i>An Introduction to Orthogonal Polynomials</i>. Gordon and Breach, New York. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-677-04150-6?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Special:BookSources/978-0-677-04150-6"><bdi>978-0-677-04150-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=An+Introduction+to+Orthogonal+Polynomials&amp;rft.pub=Gordon+and+Breach%2C+New+York&amp;rft.date=1978&amp;rft.isbn=978-0-677-04150-6&amp;rft.au=Theodore+Seio+Chihara&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAppell+sequence" class="Z3988"></span></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBedratyukLuno2020" class="citation journal cs1">Bedratyuk, L.; Luno, N. (2020). "Some Properties of Generalized Hypergeometric Appell Polynomials". <i>Carpathian Math. Publ</i>. <b>12</b> (1): <span class="nowrap">129–</span>137. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ArXiv_(identifier)?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://arxiv.org/abs/2005.01676">2005.01676</a></span>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://doi.org/10.15330%252Fcmp.12.1.129-137">10.15330/cmp.12.1.129-137</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Carpathian+Math.+Publ.&amp;rft.atitle=Some+Properties+of+Generalized+Hypergeometric+Appell+Polynomials&amp;rft.volume=12&amp;rft.issue=1&amp;rft.pages=%3Cspan+class%3D%22nowrap%22%3E129-%3C%2Fspan%3E137&amp;rft.date=2020&amp;rft_id=info%3Aarxiv%2F2005.01676&amp;rft_id=info%3Adoi%2F10.15330%2Fcmp.12.1.129-137&amp;rft.aulast=Bedratyuk&amp;rft.aufirst=L.&amp;rft.au=Luno%2C+N.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAppell+sequence" class="Z3988"></span></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="External_links">External links</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=edit&amp;section=8&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://www.encyclopediaofmath.org/index.php?title%3DAppell_polynomials">"Appell polynomials"</a>, <i><a href="https://en-m-wikipedia-org.translate.goog/wiki/Encyclopedia_of_Mathematics?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/European_Mathematical_Society?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Appell+polynomials&amp;rft.btitle=Encyclopedia+of+Mathematics&amp;rft.pub=EMS+Press&amp;rft.date=2001&amp;rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAppell_polynomials&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AAppell+sequence" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=http://mathworld.wolfram.com/AppellSequence.html">Appell Sequence</a> at <a href="https://en-m-wikipedia-org.translate.goog/wiki/MathWorld?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB" title="MathWorld">MathWorld</a></li> </ul><!-- NewPP limit report Parsed by mw‐api‐ext.codfw.main‐74795b8fcc‐dctmv Cached time: 20250117034837 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.270 seconds Real time usage: 0.454 seconds Preprocessor visited node count: 1162/1000000 Post‐expand include size: 24354/2097152 bytes Template argument size: 1014/2097152 bytes Highest expansion depth: 13/100 Expensive parser function count: 5/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 20570/5000000 bytes Lua time usage: 0.146/10.000 seconds Lua memory usage: 5820682/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 277.002 1 -total 33.10% 91.682 4 Template:Cite_journal 30.63% 84.843 1 Template:Short_description 18.56% 51.417 1 Template:Inline 16.96% 46.992 2 Template:Ambox 14.46% 40.060 2 Template:Pagetype 12.11% 33.531 5 Template:Main_other 11.29% 31.269 1 Template:SDcat 6.12% 16.940 1 Template:Springer 4.32% 11.956 1 Template:Use_mdy_dates --> <!-- Saved in parser cache with key enwiki:pcache:1649947:|#|:idhash:canonical and timestamp 20250117034837 and revision id 1228267574. Rendering was triggered because: api-parse --> </section> </div><!-- MobileFormatter took 0.022 seconds --><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --> <noscript> <img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=mobile&amp;type=1x1&amp;usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Retrieved from "<a dir="ltr" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://en.wikipedia.org/w/index.php?title%3DAppell_sequence%26oldid%3D1228267574">https://en.wikipedia.org/w/index.php?title=Appell_sequence&amp;oldid=1228267574</a>" </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Appell_sequence&amp;action=history&amp;_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"><span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="OAbot" data-user-gender="unknown" data-timestamp="1718010888"> <span>Last edited on 10 June 2024, at 09:14</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div></a> <div class="post-content footer-content"> <div id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://cs.wikipedia.org/wiki/Appellova_posloupnost" title="Appellova posloupnost – Czech" lang="cs" hreflang="cs" data-title="Appellova posloupnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://es.wikipedia.org/wiki/Serie_de_Appel" title="Serie de Appel – Spanish" lang="es" hreflang="es" data-title="Serie de Appel" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://ru.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D1%2581%25D0%25BB%25D0%25B5%25D0%25B4%25D0%25BE%25D0%25B2%25D0%25B0%25D1%2582%25D0%25B5%25D0%25BB%25D1%258C%25D0%25BD%25D0%25BE%25D1%2581%25D1%2582%25D1%258C_%25D0%2590%25D0%25BF%25D0%25BF%25D0%25B5%25D0%25BB%25D1%258F" title="Последовательность Аппеля – Russian" lang="ru" hreflang="ru" data-title="Последовательность Аппеля" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://sk.wikipedia.org/wiki/Appellova_postupnos%25C5%25A5" title="Appellova postupnosť – Slovak" lang="sk" hreflang="sk" data-title="Appellova postupnosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://sr.wikipedia.org/wiki/%25D0%2590%25D0%25BF%25D0%25B5%25D0%25BB%25D0%25BE%25D0%25B2_%25D0%25BD%25D0%25B8%25D0%25B7" title="Апелов низ – Serbian" lang="sr" hreflang="sr" data-title="Апелов низ" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://zh.wikipedia.org/wiki/%25E9%2598%25BF%25E4%25BD%25A9%25E7%2588%25BE%25E5%25BA%258F%25E5%2588%2597" title="阿佩爾序列 – Chinese" lang="zh" hreflang="zh" data-title="阿佩爾序列" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod">This page was last edited on 10 June 2024, at 09:14<span class="anonymous-show">&nbsp;(UTC)</span>.</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:About?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:General_disclaimer?_x_tr_sl=auto&amp;_x_tr_tl=en&amp;_x_tr_hl=en-GB">Disclaimers</a></li> <li id="footer-places-contact"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://stats.wikimedia.org/%23/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-terms-use"><a href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://foundation.m.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">Terms of Use</a></li> <li id="footer-places-desktop-toggle"><a id="mw-mf-display-toggle" href="https://translate.google.com/website?sl=auto&amp;tl=en&amp;hl=en-GB&amp;u=https://en.wikipedia.org/w/index.php?title%3DAppell_sequence%26mobileaction%3Dtoggle_view_desktop" data-event-name="switch_to_desktop">Desktop</a></li> </ul> </div> </footer> </div> </div> <div class="mw-notification-area" data-mw="interface"></div><!-- v:8.3.1 --> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-797994fbf8-2lrv7","wgBackendResponseTime":168,"wgPageParseReport":{"limitreport":{"cputime":"0.270","walltime":"0.454","ppvisitednodes":{"value":1162,"limit":1000000},"postexpandincludesize":{"value":24354,"limit":2097152},"templateargumentsize":{"value":1014,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":5,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":20570,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 277.002 1 -total"," 33.10% 91.682 4 Template:Cite_journal"," 30.63% 84.843 1 Template:Short_description"," 18.56% 51.417 1 Template:Inline"," 16.96% 46.992 2 Template:Ambox"," 14.46% 40.060 2 Template:Pagetype"," 12.11% 33.531 5 Template:Main_other"," 11.29% 31.269 1 Template:SDcat"," 6.12% 16.940 1 Template:Springer"," 4.32% 11.956 1 Template:Use_mdy_dates"]},"scribunto":{"limitreport-timeusage":{"value":"0.146","limit":"10.000"},"limitreport-memusage":{"value":5820682,"limit":52428800}},"cachereport":{"origin":"mw-api-ext.codfw.main-74795b8fcc-dctmv","timestamp":"20250117034837","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Appell sequence","url":"https:\/\/en.wikipedia.org\/wiki\/Appell_sequence","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1090038","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1090038","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-03-26T03:32:40Z","dateModified":"2024-06-10T09:14:48Z","headline":"type of polynomial sequence"}</script> <script>(window.NORLQ=window.NORLQ||[]).push(function(){var ns,i,p,img;ns=document.getElementsByTagName('noscript');for(i=0;i<ns.length;i++){p=ns[i].nextSibling;if(p&&p.className&&p.className.indexOf('lazy-image-placeholder')>-1){img=document.createElement('img');img.setAttribute('src',p.getAttribute('data-src'));img.setAttribute('width',p.getAttribute('data-width'));img.setAttribute('height',p.getAttribute('data-height'));img.setAttribute('alt',p.getAttribute('data-alt'));p.parentNode.replaceChild(img,p);}}});</script> <script>function gtElInit() {var lib = new google.translate.TranslateService();lib.translatePage('en', 'en', function () {});}</script> <script src="https://translate.google.com/translate_a/element.js?cb=gtElInit&amp;hl=en-GB&amp;client=wt" type="text/javascript"></script> </body> </html>

Pages: 1 2 3 4 5 6 7 8 9 10