Hypergeometric multiple orthogonal polynomials

This thesis is devoted to the analysis of multiple orthogonal polynomials for indices on the so-called step-line with respect to absolutely continuous measures on the positive real line, whose moments are given by ratios of Pochhammer symbols (also known as rising factorials). We investigate both type I and type II multiple orthogonal polynomials, though the main focus is on the type II polynomials. For the former, the characterisation includes Rodrigues-type formulas for the type I polynomials and type I functions. On the latter, the characterisation includes explicit representations as terminating generalised hypergeometric series as well as solutions of differential equations and recurrence relations, and an analysis of their asymptotic behaviour and the location of their zeros. We investigate the link of these polynomials with branched-continued-fraction representations of generalised hypergeometric series, which were introduced to solve total-positivity problems in combinatorics. The polynomials analysed here also have direct applications to the study of Painlevé equations and to random matrix theory. We give a detailed characterisation of two new families of multiple orthogonal polynomials associated with Nikishin systems of 2 absolutely continuous measures. These measures are supported on the positive real line and on the interval (0,1) and they admit integral representations via the confluent hypergeometric function of the second kind (also known as the Tricomi function) and Gauss' hypergeometric function, respectively. The vectors of orthogonality weights satisfy matrix Pearson-type differential equations, linked to the action of the differentiation operator on the type II polynomials and type I functions as a shift in their index and parameters. As a result, the type II polynomials and type I functions satisfy Hahn's property. We further draw the links between these two families of multiple orthogonal polynomials and other known polynomial sets via limiting relations or specialisations. Examples of such connections encompass the components of the cubic decomposition of Hahn-classical threefold-symmetric 2-orthogonal polynomials as well as Jacobi-Piñeiro polynomials and multiple orthogonal polynomials with respect to Macdonald functions.