New connection formulae for the q-orthogonal polynomials via a series expansion of the q-exponential

TL;DR

This paper introduces new connection formulae for q-orthogonal polynomials by expressing the q-exponential as an infinite product of ordinary exponentials, linking q-polynomials to classical counterparts.

Contribution

It presents novel nonlinear connection formulae for q-orthogonal polynomials using a series expansion of the q-exponential function.

Findings

Derived new connection formulae for q-Hermite, q-Laguerre, and q-Gegenbauer polynomials.

Established relationships between q-polynomials and classical orthogonal polynomials.

Provided a method to express q-polynomials in terms of classical analogs.

Abstract

Using a realization of the q-exponential function as an infinite multiplicative sereis of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and q-Gegenbauer polynomials in terms of their respective classical analogs.