A note on certain multiple integrals

An analytical expression is given for the multiple integral ∫⋅⋅⋅∫dμ (xn+1) dμ (xn+2) ⋅⋅⋅dμ (xN) Π1⩽j<k⩽N‖xj −xk‖β, where 0⩽n⩽N, β=1,2, or 4 and the positive measure dμ (x) is such that all its moments exist, ∫dμ (x) xj<∞, j=0,1,2,⋅⋅⋅. The case dμ (x) =dx, for −1⩽x⩽1, and dμ (x) =0, for ‖x‖≳1, is given as an example. In the limit N→∞ the correlation functions of this example, the so-called Legendre ensembles, coincide with those of the circular or the Gaussian ensembles of random matrices.