Richard P. Brent, Christian Krattenthaler and Sven Ole Warnaar

Discrete analogues of Macdonald-Mehta integrals

(50 pages)

Abstract. We consider discretisations of the Macdonald-Mehta integrals from the theory of finite reflection groups. For the classical groups, Ar-1, Br and Dr, we provide closed-form evaluations in those cases for which the Weyl denominators featuring in the summands have exponents 1 and 2. Our proofs for the exponent-1 cases rely on identities for classical group characters, while most of the formulas for the exponent-2 cases are derived from a transformation formula for elliptic hypergeometric series for the root system BCr. As a byproduct of our results, we obtain closed-form product formulas for the (ordinary and signed) enumeration of orthogonal and symplectic tableaux contained in a box.

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