THE SPECTRUM OF AN INTEGRAL OPERATOR IN WEIGHTED L2 SPACES

We find the spectrum of the inverse operator of the q-difference operator Dq,xf(x) = (f(x) − f(qx))/(x(1 − q)) on a family of weighted L2 spaces. We show that the spectrum is discrete and the eigenvalues are the reciprocals of the ze-ros of an entire function. We also derive an expansion of the eigenfunctions of the q-difference operator and its inverse in terms of big q-Jacobi polynomials. This provides a q-analogue of the expansion of a plane wave in Jacobi polynomials. The coefficients are related to little q-Jacobi polynomials, which are described and proved to be orthogonal on the spectrum of the inverse operator. Explicit representations for the little q-Jacobi polynomials are given. 1. Introduction. The q-difference operator Dq,x is defined by Dq,xf(x):