A Second Addition Formula for Continuous q-Ultraspherical Polynomials

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.KeywordsQuantum GroupAddition FormulaConnection FormulaUltraspherical PolynomialWilson PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.