An identity involving partitional generalized binomial coefficients

Define coefficients (?[lambda]) by C[lambda](Ip + Z)/C[lambda](Ip) = [Sigma]k=0l [Sigma][varkappa][set membership, variant]k ([varkappa][lambda]) C?(Z)/C?(Ip), where the C[lambda]'s are zonal polynomials in p by p matrices. It is shown that C[varkappa](Z) etr(Z)/k! = [Sigma]l=k[infinity] [Sigma][lambda][set membership, variant]l ([varkappa][lambda]) C[lambda](Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all ([varkappa][lambda]), [varkappa] [set membership, variant] k, k <= 3. Several identities involving the ([varkappa][lambda])'s are derived. These are used to derive explicit expressions for coefficients of C[lambda](Z)/l! in expansions of P(Z), etr(Z)/k! for all monomials P(Z) in sj = tr Zj of degree k <= 5.