DISCRETE ORTHOGONAL POLYNOMIAL-EXPANSIONS OF AVERAGED FUNCTIONS

Starting from the classical Fourier coefficients of a given function f(x), Boas and Izumi [J. Indian Math. Sec. 24(1960), 191-210] derived an explicit expression for the Fourier coefficients of g(x), or appropriately defined average of f(x). Later, Askey [J. Math. Anal. Appl. 14 (1966), 326-331] demonstrated how their result may be obtained more naturally as a special case of Fourier expansion in terms of Jacobi polynomials P-n((alpha,beta))(x) as the orthogonal eigenfunctions. In this paper we present an extension of this idea to classes of polynomials that satisfy an orthogonality relation with respect to a discrete rather than a continuous measure. In particular, we focus on the q-Hahn polynomials Q(n)(q(-x); a, b, N; q), defined by a terminating basic hypergeometric power series. This class has the ordinary Hahn polynomials Q(n)(a; alpha, beta, N) and the little q-Jacobi polynomials p(n)(x; alpha, beta; q) as limiting cases. Both these classes have the ordinary Jacobi polynomials as a limiting case. (C) 1995 Academic Press, Inc.