# Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Lima, Helder, Loureiro, Ana F. (2022) Multiple orthogonal polynomials with respect to Gauss' hypergeometric function. Studies in Applied Mathematics, 148 (1). pp. 154-185. ISSN 0022-2526. (doi:10.1111/sapm.12437) (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 has direct applications in the investigation of singular values of products of Ginibre random matrices 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 indices lie on the step-line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. We obtain Rodrigues-type formulas for type I polynomials and functions, while a more detailed characterization is given for the type II polynomials (aka 2-orthogonal polynomials) that include an explicit expression as a terminating hypergeometric series, a third-order differential equation, and a third-order recurrence relation. The asymptotic behavior of their recurrence coefficients mimics those of Jacobi–Piñeiro polynomials, based on which their asymptotic zero distribution and a Mehler–Heine asymptotic formula near the origin are given. Particular choices of the parameters and confluence relations give some known systems such as special cases of the Jacobi–Piñeiro polynomials, Jacobi-type 2-orthogonal polynomials, components of the cubic decomposition of threefold symmetric Hahn-classical polynomials, and multiple orthogonal polynomials with respect to confluent hypergeometric functions of the second kind.

Item Type: Article 10.1111/sapm.12437 Multiple orthogonal polynomials, Gauss hypergeometric function, Nikishin system, Rodrigues-type formula, generalised hypergeometric series, 2-orthogonal polynomials, Hahn classical Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, CalculusQ Science > QA Mathematics (inc Computing science) > QA351 Special functions Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Ana F. Loureiro 04 Jan 2021 01:06 UTC 22 Feb 2022 15:32 UTC https://kar.kent.ac.uk/id/eprint/85292 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-4137-8822