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Bilinear generating functions for orthogonal polynomials. (English) Zbl 0941.33011
Authors’ abstract: Using realizations of the positive discrete series representations of the Lie algebra \(\mathfrak{su}(1,1)\) in terms of Meixner-Pollaczek polynomials, the action of \(\mathfrak{su}(1,1)\) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result,a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner-Pollaczek polynomials ; this result is also known as the Burchnall-Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra \(\mathcal{U}_q(\mathfrak{su}(1,1))\) a similar analysis is performed and leads to a bilinear generating function for Askey-Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials.
Reviewer: R.Koekoek (Delft)

MSC:
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
17B20 Simple, semisimple, reductive (super)algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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