EXPANSION OF ANALYTIC FUNCTIONS IN q-ORTHOGONAL POLYNOMIALS MORITZ SIMON AND SERGEI K. SUSLOV Abstract. A classical result on expansion of an analytic func- tion in a series of Jacobi polynomials is extended to a class of q- orthogonal polynomials containing the fundamental Askey–Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed. 1. Main Results The representation of an analytic function as a series involving poly- nomials is a fundamental problem in classical analysis and approxima- tion theory; see for instance [9, 13] and references therein. In the pre- vious note [26] we have discussed an analog of the Cauchy–Hadamard formula, well-known for expansions in power series [1, 18, 28]. In the case of polynomial expansions f (x)= ∞ X n=0 c n p n (x) (1.1) the following theorem holds [26]. Theorem 1 (Analog of Cauchy–Hadamard formula). Let E ε be an ellipse in the complex x-plane, explicitly  Re x a ε ¶ 2 +  Im x b ε ¶ 2 =1 (1.2) with semiaxes given by a ε = 1 2 ( q ε + q -ε ) , b ε = 1 2 fl fl q ε - q -ε fl fl (1.3) Date : February 23, 2006. 1991 Mathematics Subject Classification. 33D45, 42C10; 33D15. Key words and phrases. Basic hypergeometric functions, q-orthogonal polyno- mials, Askey–Wilson polynomials, continuous q-Jacobi polynomials, continuous dual q-Hahn polynomials, continuous q-ultraspherical polynomials, Al-Salam and Chihara polynomials, continuous big q-Hermite polynomials, continuous q-Hermite polynomials, ˇ Cebyˇsev polynomials, Jacobi polynomials. 1