Multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices, and branched continued fractions

TL;DR

This paper explores the unexpected links between multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices, and branched continued fractions, extending Viennot's combinatorial theory beyond tridiagonal cases.

Contribution

It introduces a partial extension of Viennot's combinatorial framework to lower-Hessenberg production matrices that are not necessarily tridiagonal.

Findings

Established connections between multiple orthogonal polynomials and branched continued fractions.

Extended Viennot's combinatorial theory to a broader class of production matrices.

Provided new insights into the structure of $d$-orthogonal polynomials.

Abstract

I analyze an unexpected connection between multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot's combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal.