  Motivated by analogous results for the symmetric group and compact Lie
groups, we study the distribution of the number of fixed vectors of a random
element of a finite classical group. We determine the limiting moments of these
distributions, and find exactly how large the rank of the group has to be in
order for the moment to stabilize to its limiting value. The proofs require a
subtle use of some q-series identities. We also point out connections with
orthogonal polynomials.
