Anatomy of a q-generalization of the Laguerre/Hermite Orthogonal Polynomials

TL;DR

This paper explores a q-analogue of classical Laguerre and Hermite orthogonal polynomials, providing explicit formulas, algebraic structures, and differential equations, thereby extending their mathematical framework.

Contribution

It introduces a new q-generalization of these polynomials, detailing recursive coefficients, algebraic generator matrix elements, and deriving q-deformed Toda equations.

Findings

Explicit recursive coefficients derived

Matrix elements of Heisenberg algebra generators computed

q-deformed Toda equations established

Abstract

We study a q-generalization of the classical Laguerre/Hermite orthogonal polynomials. Explicit results include: the recursive coefficients, matrix elements of generators for the Heisenberg algebra, and the Hankel determinants. The power of quadratic relation is illustrated by comparing two ways of calculating recursive coefficients. Finally, we derive a q-deformed version of the Toda equations for both q-Laguerre/Hermite ensembles.