Abstract

We find the spectrum of the inverse operator of the q-difference operator D_q,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 zeros 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 coeficients 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.

Authors