Discrete analogues of Macdonald-Mehta integrals
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
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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