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Orthogonal Polynomials, Asymptotics and Heun Equations
Yang Chen1; Galina Filipuk2; Longjun Zhan1
Source PublicationarXiv

The Painlev´e equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of “classical” weights multiplied by suitable “deformation factors”, usually dependent on a “time variable” t. From ladder operators [12–14, 30] one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlev´e and related functions appear as the residues of these rational functions. We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials Pn(x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by Pn(x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equations. Heun equations are of considerable importance in mathematical physics and in the special cases they degenerate to the hypergeometric and confluent hypergeometric equations (see, for instance, [1, 23, 36]). In this paper we look at three type of weights: the Jacobi type, the Laguerre type and the weights deformed by the indicator function of (a, b) χ(a,b) and the step function θ(x). In particular, we consider the following Jacobi type weights: 1.1) x α(1 − x) β e −tx, x ∈ [0, 1], α, β, t > 0; 1.2) x α(1 − x) β e −t/x, x ∈ (0, 1], α, β, t > 0; 1.3) (1 − x 2 ) α(1 − k 2x 2 ) β , x ∈ [−1, 1], α, β > 0, k2 ∈ (0, 1); the Laguerre type weights: 2.1) x α(x + t) λ e −x , x ∈ [0,∞), t, α, λ > 0; 2.2) x αe −x−t/x, x ∈ (0,∞), α, t > 0.We also study another type of deformation when the classical weights are multiplied by χ or θ: 3.1) e−x 2 (1 − χ(−a,a) (x)), x ∈ R, a > 0; 3.2) (1 − x 2 ) α(1 − χ(−a,a) (x)), x ∈ [−1, 1], a ∈ (0, 1), α > 0; 3.3) x αe −x (A + Bθ(x − t)), x ∈ [0,∞), α, t > 0, A ≥ 0, A + B ≥ 0. The weights mentioned above were studied in a series of papers related to the deformation of “classical” weights [4, 5, 9, 10, 14, 28, 29, 32, 43].

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Document TypeJournal article
CollectionFaculty of Science and Technology
Corresponding AuthorLongjun Zhan
Affiliation1.Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China
2.Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Yang Chen,Galina Filipuk,Longjun Zhan. Orthogonal Polynomials, Asymptotics and Heun Equations[J]. arXiv,2019.
APA Yang Chen,Galina Filipuk,&Longjun Zhan.(2019).Orthogonal Polynomials, Asymptotics and Heun Equations.arXiv.
MLA Yang Chen,et al."Orthogonal Polynomials, Asymptotics and Heun Equations".arXiv (2019).
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