Bidiagonal factorization of the recurrence matrix for the Hahn multiple orthogonal polynomials

TL;DR

This paper presents a bidiagonal matrix factorization of the recurrence matrix for Hahn multiple orthogonal polynomials, expressed via hypergeometric functions, and extends it to related polynomial families within the Askey scheme.

Contribution

It introduces a new bidiagonal factorization for Hahn multiple orthogonal polynomials and their descendants, using hypergeometric functions and contiguous relations, with positivity conditions analyzed.

Findings

Bidiagonal factorization expressed in terms of ${}_3F_2$ hypergeometric functions.

Extension of the factorization to multiple polynomial families in the Askey scheme.

Identification of regions where the factorization is positive.

Abstract

This paper explores a factorization using bidiagonal matrices of the recurrence matrix of Hahn multiple orthogonal polynomials. The factorization is expressed in terms of ratios involving the generalized hypergeometric function ${}_3F_2$ and is proven using recently discovered contiguous relations. Moreover, employing the multiple Askey scheme, a bidiagonal factorization is derived for the Hahn descendants, including Jacobi-Pi\~neiro, multiple Meixner (kinds I and II), multiple Laguerre (kinds I and II), multiple Kravchuk, and multiple Charlier, all represented in terms of hypergeometric functions. For the cases of multiple Hahn, Jacobi-Pi\~neiro, Meixner of kind II, and Laguerre of kind I, where there exists a region where the recurrence matrix is nonnegative, subregions are identified where the bidiagonal factorization becomes a positive bidiagonal factorization.