The density matrix of the canonically transformed multidimensional Hamiltonian in the Fock basis

The matrix elements of the density operator exp(-tauH) are calculated exactly for Hamiltonians obtained from the N-dimensional isotropic-oscillator Hamiltonian by the canonical transformation. These matrix elements are expressed in terms of Hermite's polynomials with 2N variables. The matrix elements of the density operator for the one-dimensional case are also given, the results being expressed in terms of the Laguerre and Legendre polynomials with two variables. The characteristic distribution functions for the co-ordinates, momenta and energy operators are obtained. The expressions for the moments of these distribution functions are expressed in terms of the Hermite polynomials with many variables. New generating functions and new relations are obtained for the Laguerre and Legendre polynomials and for the Hermite polynomials with two variables