Dual addition formulas: the case of continuous $q$-ultraspherical and $q$-Hermite polynomials

TL;DR

This paper establishes the dual addition formula for continuous $q$-ultraspherical polynomials using $q$-Racah polynomials and explores the limit case of $q$-Hermite polynomials, advancing the understanding of their algebraic structure.

Contribution

It provides the first explicit dual addition formula for continuous $q$-ultraspherical polynomials and links it to $q$-Racah polynomials and the Rahman--Verma addition formula.

Findings

Dual addition formula expressed via $q$-Racah polynomials

Derived from Rahman--Verma addition formula using self-duality

Limit case analysis for continuous $q$-Hermite polynomials

Abstract

We settle the dual addition formula for continuous $q$-ultraspherical polynomials as an expansion in terms of special $q$-Racah polynomials for which the constant term is given by the linearization formula for the continuous $q$-ultraspherical polynomials. In a second proof we derive the dual addition formula from the Rahman--Verma addition formula for these polynomials by using the self-duality of the polynomials. We also consider the limit case of continuous $q$-Hermite polynomials.