  It is well known that if one integrates a Schur function indexed by a
partition $\lambda$ over the symplectic (resp. orthogonal) group, the integral
vanishes unless all parts of $\lambda$ have even multiplicity (resp. all parts
of $\lambda$ are even). In a recent paper of Rains and Vazirani, Macdonald
polynomial generalizations of these identities and several others were
developed and proved using Hecke algebra techniques. However at $q=0$ (the
Hall-Littlewood level), these approaches do not work, although one can obtain
the results by taking the appropriate limit. In this paper, we develop a direct
approach for dealing with this special case. This technique allows us to prove
some identities that were not amenable to the Hecke algebra approach, as well
as to explicitly control the nonzero values. Moreover, we are able to
generalize some of the identities by introducing extra parameters. This leads
us to a finite-dimensional analog of a recent result of Warnaar, which uses the
Rogers-Szeg\"o polynomials to unify some existing summation type formulas for
Hall-Littlewood functions.
