Strong Asymptotics of the Generating Polynomials of the Stirling Numbers of the Second Kind
Identifieur interne : 001016 ( Istex/Corpus ); précédent : 001015; suivant : 001017Strong Asymptotics of the Generating Polynomials of the Stirling Numbers of the Second Kind
Auteurs : Christian ElbertSource :
- Journal of Approximation Theory [ 0021-9045 ] ; 2001.
English descriptors
Abstract
For the horizontal generating functions Pn(z)=∑nk=1S(n, k)zk of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Qn(z)=Pn(nz) there are two main results: an oscillating asymptotic for z∈(−e, 0) and a uniform asymptotic on every compact subset of C\[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.
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DOI: 10.1006/jath.2000.3533
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By using the saddle point method for Qn(z)=Pn(nz) there are two main results: an oscillating asymptotic for z∈(−e, 0) and a uniform asymptotic on every compact subset of C\[−e, 0]. 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Szekeres</name></json:item></author><host><volume>28</volume><pages><last>498</last><first>494</first></pages><author></author><title>Ann. of Math. Statist.</title></host><title>On Borel fields over finite sets</title></json:item><json:item><host><author></author><title>Asymptotic Methods in Analysis</title></host></json:item><json:item><host><author></author><title>Advanced Combinatorics</title></host></json:item><json:item><host><author></author></host></json:item><json:item><author><json:item><name>D.S. Lubinsky</name></json:item><json:item><name>A. Sidi</name></json:item></author><host><volume>14</volume><pages><last>379</last><first>341</first></pages><author></author><title>Analysis</title></host><title>Strong asymptotics for polynomials biorthogonal to powers of logx</title></json:item><json:item><author><json:item><name>L. Moser</name></json:item><json:item><name>M. Wyman</name></json:item></author><host><volume>49</volume><pages><last>54</last><first>49</first></pages><author></author><title>Trans. Roy. Soc. Canada</title></host><title>An asymptotic formula for the Bell numbers</title></json:item><json:item><host><author></author><title>Asymptotics and Special Functions</title></host></json:item><json:item><host><author></author><title>Orthogonal Polynomials</title></host></json:item><json:item><host><author></author><title>Asymptotic Approximations of Integrals</title></host></json:item></refBibs><genre><json:string>research-article</json:string></genre><host><volume>109</volume><pii><json:string>S0021-9045(00)X0007-X</json:string></pii><pages><last>217</last><first>198</first></pages><issn><json:string>0021-9045</json:string></issn><issue>2</issue><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><title>Journal of Approximation Theory</title><publicationDate>2001</publicationDate></host><categories><wos><json:string>science</json:string><json:string>mathematics</json:string></wos><scienceMetrix><json:string>natural sciences</json:string><json:string>mathematics & statistics</json:string><json:string>numerical & computational mathematics</json:string></scienceMetrix></categories><publicationDate>2001</publicationDate><copyrightDate>2001</copyrightDate><doi><json:string>10.1006/jath.2000.3533</json:string></doi><id>2441BD616141DAF002D7BE129B87D4A58CA2C7F9</id><score>2.0657551</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/2441BD616141DAF002D7BE129B87D4A58CA2C7F9/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/2441BD616141DAF002D7BE129B87D4A58CA2C7F9/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/2441BD616141DAF002D7BE129B87D4A58CA2C7F9/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Strong Asymptotics of the Generating Polynomials of the Stirling Numbers of the Second Kind</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>ELSEVIER</publisher><availability><p>©2001 Academic Press</p></availability><date>2001</date></publicationStmt><notesStmt><note>Communicated by Doron, S. Lubinsky</note><note type="content">Section title: Regular Article</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Strong Asymptotics of the Generating Polynomials of the Stirling Numbers of the Second Kind</title><author xml:id="author-1"><persName><forename type="first">Christian</forename><surname>Elbert</surname></persName><affiliation>Department of Mathematics, University of Trier, 54286, Trier, Germany</affiliation></author></analytic><monogr><title level="j">Journal of Approximation Theory</title><title level="j" type="abbrev">YJATH</title><idno type="pISSN">0021-9045</idno><idno type="PII">S0021-9045(00)X0007-X</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="2001"></date><biblScope unit="volume">109</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="198">198</biblScope><biblScope unit="page" to="217">217</biblScope></imprint></monogr><idno type="istex">2441BD616141DAF002D7BE129B87D4A58CA2C7F9</idno><idno type="DOI">10.1006/jath.2000.3533</idno><idno type="PII">S0021-9045(00)93533-0</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2001</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>For the horizontal generating functions Pn(z)=∑nk=1S(n, k)zk of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Qn(z)=Pn(nz) there are two main results: an oscillating asymptotic for z∈(−e, 0) and a uniform asymptotic on every compact subset of C\[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.</p></abstract><textClass xml:lang="en"><keywords scheme="keyword"><list><head>Keywords</head><item><term>Stirling numbers of the second kind</term></item><item><term>generating functions</term></item><item><term>Plancherel Rotach asymptotics</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="2001">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/2441BD616141DAF002D7BE129B87D4A58CA2C7F9/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Elsevier, elements deleted: tail"><istex:xmlDeclaration>version="1.0" encoding="utf-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//ES//DTD journal article DTD version 4.5.2//EN//XML" URI="art452.dtd" name="istex:docType"></istex:docType><istex:document><converted-article version="4.5.2" docsubtype="fla" xml:lang="en"><item-info><jid>YJATH</jid><aid>93533</aid><ce:pii>S0021-9045(00)93533-0</ce:pii><ce:doi>10.1006/jath.2000.3533</ce:doi><ce:copyright type="full-transfer" year="2001">Academic Press</ce:copyright></item-info><head><ce:article-footnote><ce:label>☆</ce:label><ce:note-para>Communicated by Doron, S. Lubinsky</ce:note-para></ce:article-footnote><ce:dochead><ce:textfn>Regular Article</ce:textfn></ce:dochead><ce:title>Strong Asymptotics of the Generating Polynomials of the Stirling Numbers of the Second Kind</ce:title><ce:author-group><ce:author><ce:given-name>Christian</ce:given-name><ce:surname>Elbert</ce:surname></ce:author><ce:affiliation><ce:textfn>Department of Mathematics, University of Trier, 54286, Trier, Germany<ce:footnote id="F1"><ce:label>f1</ce:label><ce:note-para>E-mail: christian.elbert@dglux.lu</ce:note-para></ce:footnote><ce:cross-ref refid="F1"><ce:sup>f1</ce:sup></ce:cross-ref></ce:textfn></ce:affiliation></ce:author-group><ce:date-received day="1" month="9" year="1999"></ce:date-received><ce:date-accepted day="12" month="9" year="2000"></ce:date-accepted><ce:abstract><ce:section-title>Abstract</ce:section-title><ce:abstract-sec><ce:simple-para>For the horizontal generating functions <ce:italic>P<ce:inf>n</ce:inf></ce:italic>(<ce:italic>z</ce:italic>)=∑<ce:sup><ce:italic>n</ce:italic></ce:sup><ce:inf><ce:italic>k</ce:italic>=1</ce:inf><ce:hsp sp="0.25"></ce:hsp><ce:italic>S</ce:italic>(<ce:italic>n</ce:italic>, <ce:italic>k</ce:italic>)<ce:hsp sp="0.25"></ce:hsp><ce:italic>z<ce:sup>k</ce:sup></ce:italic> of the Stirling numbers of the second kind, strong asymptotics are established, as <ce:italic>n</ce:italic>→∞. By using the saddle point method for <ce:italic>Q<ce:inf>n</ce:inf></ce:italic>(<ce:italic>z</ce:italic>)=<ce:italic>P<ce:inf>n</ce:inf></ce:italic>(<ce:italic>nz</ce:italic>) there are two main results: an oscillating asymptotic for <ce:italic>z</ce:italic>∈(−<ce:italic>e</ce:italic>, 0) and a uniform asymptotic on every compact subset of <math altimg="si1.gif"><of>C</of></math>\[−<ce:italic>e</ce:italic>, 0]. Finally, an Airy asymptotic in the neighborhood of −<ce:italic>e</ce:italic> is deduced.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:keywords><ce:section-title>Keywords</ce:section-title><ce:keyword><ce:text>Stirling numbers of the second kind</ce:text></ce:keyword><ce:keyword><ce:text>generating functions</ce:text></ce:keyword><ce:keyword><ce:text>Plancherel Rotach asymptotics</ce:text></ce:keyword></ce:keywords></head></converted-article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Strong Asymptotics of the Generating Polynomials of the Stirling Numbers of the Second Kind</title></titleInfo><titleInfo type="alternative" lang="en" contentType="CDATA"><title>Strong Asymptotics of the Generating Polynomials of the Stirling Numbers of the Second Kind</title></titleInfo><name type="personal"><namePart type="given">Christian</namePart><namePart type="family">Elbert</namePart><affiliation>Department of Mathematics, University of Trier, 54286, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="Full-length article"></genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">2001</dateIssued><copyrightDate encoding="w3cdtf">2001</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">For the horizontal generating functions Pn(z)=∑nk=1S(n, k)zk of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Qn(z)=Pn(nz) there are two main results: an oscillating asymptotic for z∈(−e, 0) and a uniform asymptotic on every compact subset of C\[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.</abstract><note>Communicated by Doron, S. Lubinsky</note><note type="content">Section title: Regular Article</note><subject lang="en"><genre>Keywords</genre><topic>Stirling numbers of the second kind</topic><topic>generating functions</topic><topic>Plancherel Rotach asymptotics</topic></subject><relatedItem type="host"><titleInfo><title>Journal of Approximation Theory</title></titleInfo><titleInfo type="abbreviated"><title>YJATH</title></titleInfo><genre type="journal">journal</genre><originInfo><dateIssued encoding="w3cdtf">200104</dateIssued></originInfo><identifier type="ISSN">0021-9045</identifier><identifier type="PII">S0021-9045(00)X0007-X</identifier><part><date>200104</date><detail type="volume"><number>109</number><caption>vol.</caption></detail><detail type="issue"><number>2</number><caption>no.</caption></detail><extent unit="issue pages"><start>157</start><end>314</end></extent><extent unit="pages"><start>198</start><end>217</end></extent></part></relatedItem><identifier type="istex">2441BD616141DAF002D7BE129B87D4A58CA2C7F9</identifier><identifier type="DOI">10.1006/jath.2000.3533</identifier><identifier type="PII">S0021-9045(00)93533-0</identifier><accessCondition type="use and reproduction" contentType="copyright">©2001 Academic Press</accessCondition><recordInfo><recordContentSource>ELSEVIER</recordContentSource><recordOrigin>Academic Press, ©2001</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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