# Bilinear summation formulas from quantum algebra representations

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## Abstract

Summary: 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 $2\varphi_1$-series, which can be considered as a special case of the dual transmutation kernel for little $q$-Jacobi functions.
Original language Undefined/Unknown 383-416 34 The Ramanujan Journal: an international journal devoted to areas of mathematics influenced by Ramanujan 8 3 Published - 2004