Discrete Orthogonal Polynomial Expansions of Averaged Functions

Starting from the classical Fourier coefficients of a given function ƒ(x), Boas and Izumi [J. Indian Math. Soc.24 (1960), 191-210] derived an explicit expression for the Fourier coefficients of g(x), or appropriately defined average of ƒ(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(α,β)n(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 Qn(q−x; a, b, N; q), defined by a terminating basic hypergeometric power series. This class has the ordinary Hahn polynomials Qn(x; α, β, N) and the little q-Jacobi polynomials pn(x; α, β; q) as limiting cases. Both these classes have the ordinary Jacobi polynomials as a limiting case.