Moments of the characteristic polynomial in the three ensembles of random matrices

Moments of the characteristic polynomial of a random matrix taken from any of the three ensembles, orthogonal, unitary or symplectic, are given either as a determinant or a Pfaffian or as a sum of determinants. For Gaussian ensembles, on comparing two expressions of the same moment one obtains two remarkable identities, one between an nxn determinant and an mxm determinant and another between the Pfaffian of a 2nx2n anti-symmetric matrix and a sum of mxm determinants. (author)