# Dual addition formula for continuous $q$-ultraspherical polynomials

Research paper by Tom H. Koornwinder

Indexed on: 26 Mar '18Published on: 26 Mar '18Published in: arXiv - Mathematics - Classical Analysis and ODEs

#### 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. The paper starts with a tutorial on duality properties of orthogonal polynomials in the ($q$-)Askey scheme.