Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

TL;DR

This paper introduces new multiple orthogonal polynomials related to Gauss' hypergeometric function, exploring their properties, differential equations, asymptotics, and connections to known polynomial systems and applications in random matrix theory and Painlevé equations.

Contribution

It provides a novel class of multiple orthogonal polynomials with explicit formulas, differential equations, and asymptotic analysis, expanding the theory and applications of hypergeometric-based polynomials.

Findings

Explicit Rodrigues-type formula for type I polynomials

Third-order differential and recurrence relations for type II polynomials

Asymptotic zero distribution and Mehler-Heine formula derived

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 Painlev\'e 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-Pi\~neiro 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-Pi\~neiro 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.