New connection formulae for some q-orthogonal polynomials in q-Askey scheme

TL;DR

This paper derives new nonlinear connection formulae linking q-orthogonal polynomials like continuous q-Laguerre and q-Gegenbauer to their classical counterparts, using a novel realization of the q-exponential function.

Contribution

It introduces new connection formulae for q-orthogonal polynomials based on a special realization of the q-exponential function, expanding the understanding of their relationships.

Findings

Derived nonlinear connection formulae for multiple q-orthogonal polynomials.

Expressed q-polynomials in terms of classical analogues using q-exponential series.

Enhanced methods for relating q-polynomials to classical polynomials.

Abstract

New nonlinear connection formulae of the q-orthogonal polynomials, such continuous q-Laguerre, continuous big q-Hermite, q-Meixner-Pollaczek and q-Gegenbauer polynomials, in terms of their respective classical analogues are obtained using a special realization of the q-exponential function as infinite multiplicative series of ordinary exponential function.