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Multivariable orthogonal polynomials and coupling coefficients for discrete series representations. (English) Zbl 0946.33013
The paper provides a unified group theoretical interpretation for many orthogonal and biorthogonal polynomials which are the multivariable generalizations of Hahn, Jacobi and continuous Hahn polynomials. The polynomials arise as coupling coefficients (generalized Clebsch-Gordon coefficients) for the tensor products $${\mathcal A}^{\nu_1}\otimes\cdots\otimes{\mathcal A}^{\nu_n}$$ of analytic continuations $${\mathcal A}^\nu(\nu>0)$$ of the holomorphic discrete series representations of the group $$SU(1,1)$$. The case $$n=2$$ has been previously treated by J. Peetre [Bull. Sci. Math. 116, 265-284 (1992; Zbl 0764.30008)]. The geometric interpretation leads to convolution formulas in a simple and unified method. Also another family of multivariable polynomials is introduced and studied. The polynomials in this family, called coupling kernels, are connected with the projections of $${\mathcal A}^{\nu_1}\otimes\cdots\otimes{\mathcal A}^{\nu_n}$$ onto its isotypic subspaces. They occur as reproducing kernels for spaces of coupling coefficients. Several linearization formulas involving the coupling kernels are obtained. The final section of the paper extends the theory to the Heisenberg group. This yields multivariable generalizations of Krawtchouk and Hermite polynomials.

##### MSC:
 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 33C55 Spherical harmonics 33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
##### Keywords:
Krawtchouk polynomials
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