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The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials
Min Chen1; Yang Chen2; En‐Gui Fan1
2019
Source PublicationStudies in Applied Mathematics
ISSN0022-2526
Abstract

In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight

urn:x-wiley:00222526:media:sapm12259:sapm12259-math-0001

The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near urn:x-wiley:00222526:media:sapm12259:sapm12259-math-0002, the uniform asymptotic expansion involves Airy function as urn:x-wiley:00222526:media:sapm12259:sapm12259-math-0003, and Bessel function of order α as urn:x-wiley:00222526:media:sapm12259:sapm12259-math-0004 in the neighborhood of urn:x-wiley:00222526:media:sapm12259:sapm12259-math-0005, the uniform asymptotic expansion is associated with Bessel function of order β as urn:x-wiley:00222526:media:sapm12259:sapm12259-math-0006. The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method.

KeywordAsymptotic Analysis Mathematical Physics
DOIhttps://doi.org/10.1111/sapm.12259
Language英语
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Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorEn‐Gui Fan
Affiliation1.School of Mathematical Science, FudanUniversity, Shanghai, P.R. China
2.Faculty of Science and Technology,Department of Mathematics, University ofMacau, Taipa, Macau, China
Recommended Citation
GB/T 7714
Min Chen,Yang Chen,En‐Gui Fan. The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials[J]. Studies in Applied Mathematics,2019.
APA Min Chen,Yang Chen,&En‐Gui Fan.(2019).The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials.Studies in Applied Mathematics.
MLA Min Chen,et al."The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials".Studies in Applied Mathematics (2019).
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