Bilinear Summation Formulas from Quantum Algebra Representations

Abstract

The tensor product of a positive and a negative discrete series representation of the quantum algebra U _q (su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q⁻¹, two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas involve Askey-Wilson polynomials, continuous dual q-Hahn polynomials and little q-Jacobi functions. The realization of the discrete series as q-difference operators on the spaces of holomorphic and anti-holomorphic functions, leads to a bilinear generating function for a certain type of ₂φ₁-series, which can be considered as a special case of the dual transmutation kernel for little q-Jacobi functions.