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

v2, the written version of the 2017 R.P. Agarwal Memorial Lecture, has been split into two parts. This version, v3, 17 pp., is the second part. Remark 4.2 and Section 5 are new

References in corpus (3)

• Matrix-valued orthogonal polynomials related to the quantum analogue of $(SU(2)\times SU(2),\text{diag})$
• Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions
• Dual addition formulas associated with dual product formulas

Cited by in corpus (1)

• A short survey of duality in special functions