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arXiv · 1601.06536

Discrete analogues of Macdonald-Mehta integrals

Richard P. Brent·Christian Krattenthaler·S. Ole Warnaar
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Abstract

We consider discretisations of the Macdonald--Mehta integrals from the theory of finite reflection groups. For the classical groups, $\mathrm{A}_{r-1}$, $\mathrm{B}_r$ and $\mathrm{D}_r$, 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 $\mathrm{BC}_r$. 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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Richard P. Brent, Christian Krattenthaler, S. Ole Warnaar. 2016-01-25. Discrete analogues of Macdonald-Mehta integrals. https://doi.org/10.1016/j.jcta.2016.06.005

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