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 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 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.