Bilinear summation formulas from quantum algebra representations
摘要
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.
引用
@article{arxiv.math/0201272,
title = {Bilinear summation formulas from quantum algebra representations},
author = {Wolter Groenevelt},
journal= {arXiv preprint arXiv:math/0201272},
year = {2007}
}
备注
27 pages, 1 figure