Meixner polynomials of the second kind and quantum algebras representing su(1,1)

TL;DR

This paper uses combinatorial theory to connect Meixner and Laguerre polynomials with representations of su(1,1), providing new insights into eigenvector expressions and polynomial moments.

Contribution

It introduces a combinatorial framework to relate orthogonal polynomials with su(1,1) representations, generalizing previous results and explaining polynomial eigenvector expressions.

Findings

Linking Meixner and Laguerre polynomials to su(1,1) representations.

Generalization of results on matrix entries in operator powers.

Explanation of eigenvector expressions via polynomial substitutions.

Abstract

We show how Viennot's combinatorial theory of orthogonal polynomials may be used to generalize some recent results of Sukumar and Hodges on the matrix entries in powers of certain operators in a representation of su(1,1). Our results link these calculations to finding the moments and inverse polynomial coefficients of certain Laguerre polynomials and Meixner polynomials of the second kind. As an immediate consequence of results by Koelink, Groenevelt and Van Der Jeugt, for the related operators, substitutions into essentially the same Laguerre polynomials and Meixner polynomials of the second kind may be used to express their eigenvectors. Our combinatorial approach explains and generalizes this "coincidence".