Hahn multiple orthogonal polynomials of type I: Hypergeometrical expressions

TL;DR

This paper derives explicit hypergeometric series expressions for Hahn multiple orthogonal polynomials of type I, establishes their orthogonality relations, and completes part of the Askey scheme for these polynomials.

Contribution

It provides new explicit hypergeometric formulas for Hahn and related multiple orthogonal polynomials of type I, advancing the understanding of their structure and relations.

Findings

Explicit hypergeometric expressions for Hahn type I polynomials.

Orthogonal and biorthogonal relations proven for these polynomials.

Extension of the Askey scheme for multiple orthogonal polynomials.

Abstract

Explicit expressions for the Hahn multiple polynomials of type I, in terms of Kamp\'e de F\'eriet hypergeometric series, are given. Orthogonal and biorthogonal relations are proven. Then, part of the Askey scheme for multiple orthogonal polynomials type I is completed. In particular, explicit expressions in terms of generalized hypergeometric series and Kamp\'e de F\'eriet hypergeometric series, are given for the multiple orthogonal polynomials of type I for the Jacobi-Pi\~neiro, Meixner I, Meixner II, Kravchuk, Laguerre I, Laguerre II and Charlier families.