  We study symmetric polynomials whose variables are odd-numbered Jucys-Murphy
elements. They define elements of the Hecke algebra associated to the Gelfand
pair of the symmetric group with the hyperoctahedral group. We evaluate their
expansions in zonal spherical functions and in double coset sums. These
evaluations are related to integrals of polynomial functions over orthogonal
groups. Furthermore, we give an extension of them, based on Jack polynomials.
