Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight
We study the Hankel determinants associated with the weight w(x;t)=(1−x2)β(t2−x2)αh(x),x∈(−1,1),where β>−1, α+β>−1, t>1, h(x) is analytic in a domain containing [ − 1, 1] and h(x)>0 for x∈[−1,1]. In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the dou...
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| Veröffentlicht in: | Studies in applied mathematics (Cambridge) 2015-11, Vol.135 (4), p.347-376 |
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| creator | Zeng, Zhao-Yun Xu, Shuai-Xia Zhao, Yu-Qiu |
| description | We study the Hankel determinants associated with the weight
w(x;t)=(1−x2)β(t2−x2)αh(x),x∈(−1,1),where β>−1, α+β>−1, t>1, h(x) is analytic in a domain containing [ − 1, 1] and h(x)>0 for x∈[−1,1]. In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as n→∞ and t→1. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ‐function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained. |
| doi_str_mv | 10.1111/sapm.12090 |
| format | Article |
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w(x;t)=(1−x2)β(t2−x2)αh(x),x∈(−1,1),where β>−1, α+β>−1, t>1, h(x) is analytic in a domain containing [ − 1, 1] and h(x)>0 for x∈[−1,1]. In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as n→∞ and t→1. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ‐function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.</description><identifier>ISSN: 0022-2526</identifier><identifier>EISSN: 1467-9590</identifier><identifier>DOI: 10.1111/sapm.12090</identifier><language>eng</language><publisher>Cambridge: Blackwell Publishing Ltd</publisher><subject>Applied mathematics ; Mathematical models ; Polynomials ; Studies</subject><ispartof>Studies in applied mathematics (Cambridge), 2015-11, Vol.135 (4), p.347-376</ispartof><rights>2015 Wiley Periodicals, Inc., A Wiley Company</rights><rights>2015 Massachusetts Institute of Technology</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c3750-ea8be15a6db7ea8e9b057cad086be4bc956b9d39441f56153bf470340d36f1013</citedby><cites>FETCH-LOGICAL-c3750-ea8be15a6db7ea8e9b057cad086be4bc956b9d39441f56153bf470340d36f1013</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://onlinelibrary.wiley.com/doi/pdf/10.1111%2Fsapm.12090$$EPDF$$P50$$Gwiley$$H</linktopdf><linktohtml>$$Uhttps://onlinelibrary.wiley.com/doi/full/10.1111%2Fsapm.12090$$EHTML$$P50$$Gwiley$$H</linktohtml><link.rule.ids>314,776,780,1411,27903,27904,45553,45554</link.rule.ids></links><search><creatorcontrib>Zeng, Zhao-Yun</creatorcontrib><creatorcontrib>Xu, Shuai-Xia</creatorcontrib><creatorcontrib>Zhao, Yu-Qiu</creatorcontrib><title>Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight</title><title>Studies in applied mathematics (Cambridge)</title><addtitle>Studies in Applied Mathematics</addtitle><description>We study the Hankel determinants associated with the weight
w(x;t)=(1−x2)β(t2−x2)αh(x),x∈(−1,1),where β>−1, α+β>−1, t>1, h(x) is analytic in a domain containing [ − 1, 1] and h(x)>0 for x∈[−1,1]. In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as n→∞ and t→1. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ‐function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.</description><subject>Applied mathematics</subject><subject>Mathematical models</subject><subject>Polynomials</subject><subject>Studies</subject><issn>0022-2526</issn><issn>1467-9590</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><recordid>eNp9kEtOwzAQhi0EEqWw4QSW2CGljJM4rpcVjzaolEo8urTsxAG3eWGnQI_EObgYKQGWzGZm8f0zmg-hYwID0taZk3UxID5w2EE9EkbM45TDLuoB-L7nUz_aRwfOLQGAMAo9NJtLU-b69fMDx3GMR25T1E3VmMThKsMTWa50ji90o21hSlk2DmeVxRLPtW3WVukUX8ukUgYvtHl6bg7RXiZzp49-eh89XF3en0-86e04Ph9NvSRoz3paDpUmVEapYu2suQLKEpnCMFI6VAmnkeJpwMOQZDQiNFBZyCAIIQ2ijAAJ-uik21vb6mWtXSOW1dqW7UlBmO8PWwVsS512VGIr56zORG1NIe1GEBBbX2LrS3z7amHSwW8m15t_SHE3mt_8ZrwuY1yj3_8y0q5ExNpPxWI2Fi0cTBl_FDT4AiGWe_Q</recordid><startdate>201511</startdate><enddate>201511</enddate><creator>Zeng, Zhao-Yun</creator><creator>Xu, Shuai-Xia</creator><creator>Zhao, Yu-Qiu</creator><general>Blackwell Publishing Ltd</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope></search><sort><creationdate>201511</creationdate><title>Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight</title><author>Zeng, Zhao-Yun ; Xu, Shuai-Xia ; Zhao, Yu-Qiu</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c3750-ea8be15a6db7ea8e9b057cad086be4bc956b9d39441f56153bf470340d36f1013</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Applied mathematics</topic><topic>Mathematical models</topic><topic>Polynomials</topic><topic>Studies</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Zeng, Zhao-Yun</creatorcontrib><creatorcontrib>Xu, Shuai-Xia</creatorcontrib><creatorcontrib>Zhao, Yu-Qiu</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><jtitle>Studies in applied mathematics (Cambridge)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Zeng, Zhao-Yun</au><au>Xu, Shuai-Xia</au><au>Zhao, Yu-Qiu</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight</atitle><jtitle>Studies in applied mathematics (Cambridge)</jtitle><addtitle>Studies in Applied Mathematics</addtitle><date>2015-11</date><risdate>2015</risdate><volume>135</volume><issue>4</issue><spage>347</spage><epage>376</epage><pages>347-376</pages><issn>0022-2526</issn><eissn>1467-9590</eissn><abstract>We study the Hankel determinants associated with the weight
w(x;t)=(1−x2)β(t2−x2)αh(x),x∈(−1,1),where β>−1, α+β>−1, t>1, h(x) is analytic in a domain containing [ − 1, 1] and h(x)>0 for x∈[−1,1]. In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as n→∞ and t→1. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ‐function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.</abstract><cop>Cambridge</cop><pub>Blackwell Publishing Ltd</pub><doi>10.1111/sapm.12090</doi><tpages>30</tpages><oa>free_for_read</oa></addata></record> |
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| subjects | Applied mathematics Mathematical models Polynomials Studies |
| title | Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight |
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