The spectrum of an integral operator in weighted L-2 spaces

We find the spectrum of the inverse operator of the q-difference operator D -q,D-x f (x) = (f (x) - f (qx))/(x (1-q)) on a family of weighted L-2 space s. We show that the spectrum is discrete and the eigenvalues are the recipr ocals of the zeros of an entire function. We also derive an expansion of th e eigenfunctions of the q-difference operator and its inverse in terms of b ig q-Jacobi polynomials. This provides a q-analogue of the expansion of a p lane wave in Jacobi polynomials. The coefficients are related to little q-J acobi polynomials, which are described and proved to be orthogonal on the s pectrum of the inverse operator. Explicit representations for the little q- Jacobi polynomials are given.