Theory and Applications of Special Functions pp 339-360 | Cite as

# A Second Addition Formula for Continuous *q*-Ultraspherical Polynomials

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## Abstract

This paper provides the details of Remark 5.4 in the author's paper “Askey-Wilson polynomials as zonal spherical functions on the *SU*(2) quantum group,” SIAM J. Math. Anal. **24** (1993), 795–813. In formula (5.9) of the 1993 paper a two-parameter class of Askey-Wilson polynomials was expanded as a finite Fourier series with a product of two 3ϕ2's as Fourier coefficients. The proof given there used the quantum group interpretation. Here this identity will be generalized to a 3-parameter class of Askey-Wilson polynomials being expanded in terms of continuous *q*-ultraspherical polynomials with a product of two 2ϕ2's as coefficients, and an analytic proof will be given for it. Then Gegenbauer's addition formula for ultraspherical polynomials and Rahman's addition formula for *q*-Bessel functions will be obtained as limit cases. This *q*-analogue of Gegenbauer's addition formula is quite different from the addition formula for continuous *q*-ultraspherical polynomials obtained by Rahman and Verma in 1986. Furthermore, the functions occurring as factors in the expansion coefficients will be interpreted as a special case of a system of biorthogonal rational functions with respect to the Askey-Roy *q*-beta measure. A degenerate case of this biorthogonality are Pastro's biorthogonal polynomials associated with the Stieltjes-Wigert polynomials.

## Keywords

Quantum Group Addition Formula Connection Formula Ultraspherical Polynomial Wilson Polynomial## Preview

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