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

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