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Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Lima, Helder, Loureiro, Ana F. (2020) Multiple orthogonal polynomials with respect to Gauss' hypergeometric function. arXiv, . (Submitted) (KAR id:85292)

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Abstract

A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0, 1) is studied. This type of polynomials have direct applications in the investigation of singular values of products of Ginibre matrices, in the analysis of rational solutions to Painlevé equations and are connected with branched continued fractions and total positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indexes lie on the step line, for which it is shown that differentiation on the variable gives a shift on the parameters, therefore satisfying Hahn's property. We obtain a Rodrigues-type formula for type I, while a more detailed characterisation is given for the type II polynomials (aka 2-orthogonal polynomials) which include: an explicit expression as a terminating hypergeometric series, a third-order differential equation, and a third-order recurrence relation. The asymptotic behaviour of their recurrence coefficients mimics those of Jacobi-Piñeiro polynomials, based on which, their zero asymptotic distribution and a Mehler-Heine asymptotic formula near the origin are given. Particular choices on the parameters degenerate in some known systems such as special cases of the Jacobi-Piñeiro polynomials, Jacobi-type 2-orthogonal polynomials, and components of the cubic decomposition of threefold symmetric Hahn-classical polynomials. Equally considered are confluence relations to other known polynomial sets, such as multiple orthogonal polynomials with respect to Tricomi functions.

Item Type: Article
Uncontrolled keywords: Multiple orthogonal polynomials, Gauss hypergeometric function, Nikishin system, Rodrigues-type formula, generalised hypergeometric series, 2-orthogonal polynomials, Hahn classical
Subjects: Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus
Q Science > QA Mathematics (inc Computing science) > QA351 Special functions
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Ana F. Loureiro
Date Deposited: 04 Jan 2021 01:06 UTC
Last Modified: 16 Feb 2021 14:17 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/85292 (The current URI for this page, for reference purposes)
Loureiro, Ana F.: https://orcid.org/0000-0002-4137-8822
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