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Asymptotic relations for semi-classical Laguerre orthogonal polynomials and the associated Hankel determinants
Pengju Han1; Yang Chen2
2022-08
Source PublicationJournal of mathematical physics
ISSN0022-2488
Abstract

We study the recurrence coefficients of semi-classical Laguerre orthogonal polynomials and the associated Hankel determinant generated by a semi-classical Laguerre weight w(x,t) = x α e −x−tx2 , x ∈ (0,∞), α > 0, t ≥ 0. If t = 0, it is reduced to the classical Laguerre weight. For t > 0, this weight tends to zero faster than the classical Laguerre weight as x → ∞. In the finite n dimensional case, we obtain two auxiliary quantities Rn(t) and rn(t) by using the Ladder operator approach. We show that the Hankel determinant has an integral representation in terms of Rn(t), where the quantity Rn(t) is closely related a second-order nonlinear differential equation. Furthermore, we derive a second-order nonlinear differential equation and also a second-order differential equation for the auxiliary quantity σn(t) = −∑ n−1 j=0 Rj(t), which is also related with the logarithmic derivative of the Hankel determinant. In infinite n dimensional case, we consider the asymptotic behaviors of the recurrence coefficients and the sacled Laguerre orthogonal polynomials by using the Coulomb fluid method.

KeywordRandom Matrix Theory Hankel Determinant Semi-classical Laguerre Weight Ladder Operators Orthogonal Polynomials
DOI10.1063/5.0072813
Language英語English
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Document TypeJournal article
CollectionFaculty of Science and Technology
DEPARTMENT OF MATHEMATICS
Affiliation1.Department of Mathematics and Statistics, School of Sciences, Huazhong Agriculture University, Wuhan 430072, China
2.Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China
Recommended Citation
GB/T 7714
Pengju Han,Yang Chen. Asymptotic relations for semi-classical Laguerre orthogonal polynomials and the associated Hankel determinants[J]. Journal of mathematical physics,2022.
APA Pengju Han,&Yang Chen.(2022).Asymptotic relations for semi-classical Laguerre orthogonal polynomials and the associated Hankel determinants.Journal of mathematical physics.
MLA Pengju Han,et al."Asymptotic relations for semi-classical Laguerre orthogonal polynomials and the associated Hankel determinants".Journal of mathematical physics (2022).
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