Laurent Biorthogonal Polynomials and Riordan Arrays

TL;DR

This paper explores the relationship between Laurent biorthogonal polynomials with constant recurrence coefficients and Riordan arrays, introducing a connection to orthogonal and generalized orthogonal polynomials.

Contribution

It establishes a novel link between Laurent biorthogonal polynomials and Riordan arrays, and extends the framework to generalized orthogonal polynomials with parameter-dependent recurrence.

Findings

Laurent biorthogonal polynomials with constant recurrence coefficients form Riordan arrays.

A natural association between these polynomials and orthogonal polynomials is developed.

Results are extended to generalized orthogonal polynomials depending on three parameters.

Abstract

We show that Laurent biorthogonal polynomials whose defining three-term recurrence have constant coefficients have coefficient arrays that are Riordan arrays. For each such family of Laurent biorthogonal polynomials we associate in a natural way a family of orthogonal polynomials. We also extend these results to a notion of generalized orthogonal polynomials whose recurrence depends on three parameters.