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Semi-classical Jacobi polynomials, Hankel determinants and asymptotics
Min, Chao1; Chen, Yang2
Source PublicationAnalysis and Mathematical Physics

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
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Indexed BySCIE
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000720411600001
Scopus ID2-s2.0-85119405537
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Document TypeJournal article
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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