Bilinear summation formulas from quantum algebra representations

Details Bilinear summation formulas from quantum algebra representations Bilinear summation formulas from quantum algebra representations

2002-01-29

arxiv.org/abs/math/0201272v1

Ramanujan J. 8 (2004), no. 3, 383-416

Mathematics

Quantum Algebra

27 pages, 1 figure

Scientific paper

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-phi-1 -series, which can be considered as a special case of the dual transmutation kernel for little q-Jacobi functions.

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