Character expansions for the orthogonal and symplectic groups

Formulas for the expansion of arbitrary invariant group functions in terms of the characters for the $Sp(2N),$ $SO(2N+1),$ and $SO(2N)$ groups are derived using a combinatorial method. The method is similar to one used by Balantekin to expand group functions over the characters of the $U(N)$ group. All three expansions have been checked for all $N$ by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions. An expansion of the exponential of the traces of group elements, appearing in the finite-volume gauge field partition functions, is worked out for the orthogonal and symplectic groups.