THE BIG q-JACOBI FUNCTION TRANSFORM

Abstract. We give a detailed description of the resolution of the identity of a second order q-difference operator considered as an unbounded self-adjoint operator on two different Hilbert spaces. The q-difference operator and the two choices of Hilbert spaces naturally arise from harmonic analysis on the quantum group SUq(1, 1) and SUq(2). The spectral analysis associated to SUq(1, 1) leads to the big q-Jacobi function transform together with its Plancherel measure and inversion formula. The dual orthogonality relations give a oneparameter family of non-extremal orthogonality measures for the continuous dual q-Hahn polynomials with q > 1, and explicit sets of functions which complement these polynomials to orthogonal bases of the associated Hilbert spaces. The spectral analysis associated to SUq(2) leads to a functional analytic proof of the orthogonality relations and quadratic norm evaluations for the big q-Jacobi polynomials.