A generating function for averages over the orthogonal group

The integral, $\int_{\germ{H}}$ exp (tr H$^{\prime}$X) dV(H), over the group $\germ{H}$ of orthogonal matrices with respect to the invariant measure V(H), is obtained. X is an n $\times $ n matrix, and H an n $\times $ n orthogonal matrix. Various applications of it are discussed, namely, to the non-central Wishart distribution and as a generating function for the integrals and averages of polynomials over the orthogonal group.