Integral and hypergeometric representations for multiple orthogonal polynomials

Abstract

This paper addresses two primary objectives in the realm of classical multiple orthogonal polynomials with an arbitrary number of weights. Firstly, it establishes new and explicit hypergeometric expressions for type I Hahn multiple orthogonal polynomials. Secondly, applying the residue theorem and the Mellin transform, the paper derives contour integral representations for several families of orthogonal polynomials. Specifically, it presents contour integral formulas for both type I and type II multiple orthogonal polynomials in the Laguerre of the first kind, Jacobi-Pi\~neiro, and Hahn families. The evaluation of these integrals leads to explicit hypergeometric representations.