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In our previous papers we studied Laguerre functions and polynomials on symmetric cones Ω= H/L. The Laguerre functions ℓnv, n ∈, form an orthogonal basis in L2(Ω, dμv)L and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations (πv, Hv) of the automorphism group G corresponding to a tube domain T(Ω). In this article, we consider the case where Ω is the space of positive definite Hermitian matrices and G = SU(n, n). We describe the Lie algebraic realization of πv acting in L 2(Ω, dμv) and use that to determine explicit differential equations and recurrence relations for the Laguerre functions.

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Integral Transforms and Special Functions

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