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, A_r−1, B_r and 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 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.