Integral and hypergeometric representations for multiple orthogonal polynomials

TL;DR

This paper develops new hypergeometric and contour integral representations for multiple orthogonal polynomials, enhancing analytical tools for classical families like Hahn, Laguerre, and Jacobi-Piñeiro.

Contribution

It introduces explicit hypergeometric formulas for type I Hahn polynomials and contour integral representations for multiple orthogonal polynomial families.

Findings

Derived explicit hypergeometric expressions for type I Hahn polynomials.

Established contour integral formulas for Laguerre, Jacobi-Piñeiro, and Hahn families.

Connected integral representations to hypergeometric functions.

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.