Bidiagonal matrix factorisations associated with symmetric multiple orthogonal polynomials and lattice paths

TL;DR

This paper explores the connection between bidiagonal matrix factorizations, symmetric multiple orthogonal polynomials, and lattice paths, revealing new structural insights and explicit formulas for these polynomials and their measures.

Contribution

It introduces novel links between Hessenberg matrices, multiple orthogonal polynomials, and lattice paths, expanding understanding of their interrelations and explicit representations.

Findings

Hessenberg matrices are associated with symmetric multiple orthogonal polynomials.

These matrices serve as production matrices for generating polynomials of r-Dyck paths.

Explicit hypergeometric formulas and Meijer G-function representations are provided for specific polynomial sequences.

Abstract

The central object of study in this paper are infinite banded Hessenberg matrices admitting factorisations as products of bidiagonal matrices. In the two main novel results of this paper, we show that these Hessenberg matrices are associated with the decomposition of $(r+1)$-fold symmetric $r$-orthogonal polynomials and are the production matrices of the generating polynomials of $r$-Dyck paths. We combine the aforementioned bidiagonal matrix factorisations and the recently found connection of multiple orthogonal polynomials with lattice paths and branched continued fractions to study $(r+1)$-fold symmetric $r$-orthogonal polynomials on a star-like set of the complex plane and their decomposition via multiple orthogonal polynomials on the positive real line. As an explicit example, we give formulas as terminating hypergeometric series for the Appell sequences of $(r+1)$-fold symmetric $r$-orthogonal polynomials on a star-like set and show that the densities of their orthogonality measures can be expressed via Meijer G-functions on the positive real line.