Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

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 a differentiation on the variable gives a shift on the parameters, therefore satisfying the Hahn's property. We obtain a Rodrigues-type formula for the type I, while a more detailed characterisation is given for the type II polynomials (aka 2-orthogonal polynomials) which includes: 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.

Hypergeometric Multiple Orthogonal Polynomials and Random Walks

“…In this paper we perform such a study and show that the Lima and Loureiro hypergeometric multiple orthogonal polynomials [39] are indeed random walk polynomials. For that aim we explicitly compute the value at unity of the type I linear forms.…”

“…Within this introduction we remind the reader the main aspects of multiple orthogonal polynomials relevant for our objectives. Then we give a brief description of the results in [39]. We finish the introduction by recalling some of our results in [15].…”

“…The large 𝑛 limit of the corresponding Jacobi matrix leads to a particular interesting case with a Toeplitz matrix describing the transitions. At the time we finished the writing of [15] a new paper appeared [39], in where new multiple orthogonal polynomials based on Gauss hypergeometric function weights were given. The corresponding Jacobi matrix is bounded and nonnegative.…”

“…Then, whenever the zeros of the type II multiple orthogonal polynomials are confined in the interior of the support of the weight system, the type II multiple orthogonal polynomials at unity are also positive and our method in [15] is applicable. Notice that this confinement of zeros happens for example for algebraic Chebyshev (AT) systems of weights, in particular for Nikishin systems as the hypergeometric system of weights is in some region of parameters, see [39].…”

“…Section 2 is devoted to ratio asymptotics. In Theorem 7 we use the Rodrigues type formula of [39] to get the type I linear forms at unity explicitly. This is an important finding, allowing explicit expressions for the type I stochastic matrices.…”