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Semi-classical Jacobi polynomials, Hankel determinants and asymptotics
Min, Chao1; Chen, Yang2
2022-02-01
Source PublicationAnalysis and Mathematical Physics
ISSN1664-2368
Volume12Issue:1
Abstract

We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient β(t) and the sub-leading coefficient p (n, t) of the monic orthogonal polynomials. This enables us to obtain the large n asymptotics of β(t) and p (n, t) based on the result of Kuijlaars et al. [Adv. Math. 188 (2004) 337–398]. In addition, we show the second-order differential equation satisfied by the orthogonal polynomials, with all the coefficients expressed in terms of β(t). From the t evolution of the auxiliary quantities, we prove that β(t) satisfies a second-order differential equation and R(t) = 2 n+ 1 + 2 α- 2 t(β(t) + β(t)) satisfies a particular Painlevé V equation under a simple transformation. Furthermore, we show that the logarithmic derivative of the associated Hankel determinant satisfies both the second-order differential and difference equations. The large n asymptotics of the Hankel determinant is derived from its integral representation in terms of β(t) and p (n, t).

KeywordAsymptotic Expansions Differential And Difference Equations Hankel Determinants Ladder Operators Painlevé v Semi-classical Jacobi Polynomials
DOI10.1007/s13324-021-00619-9
URLView the original
Indexed BySCIE
Language英語English
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000720411600001
Scopus ID2-s2.0-85119405537
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Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorMin, Chao
Affiliation1.School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, China
2.Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao
Recommended Citation
GB/T 7714
Min, Chao,Chen, Yang. Semi-classical Jacobi polynomials, Hankel determinants and asymptotics[J]. Analysis and Mathematical Physics,2022,12(1).
APA Min, Chao,&Chen, Yang.(2022).Semi-classical Jacobi polynomials, Hankel determinants and asymptotics.Analysis and Mathematical Physics,12(1).
MLA Min, Chao,et al."Semi-classical Jacobi polynomials, Hankel determinants and asymptotics".Analysis and Mathematical Physics 12.1(2022).
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