Differential operators associated with zonal polynomials. I

Associated with each zonal polynomial, C _k(S), of a symmetric matrix S, we define a differential operator ∂_k, having the basic property that ∂_kC_λδ_kλ, δ being Kronecker's delta, whenever κ and λ are partitions of the non-negative integer k. Using these operators, we solve the problems of determining the coefficients in the expansion of (i) the product of two zonal polynomials as a series of zonal polynomials, and (ii) the zonal polynomial of the direct sum, S⊕T, of two symmetric matrices S and T, in terms of the zonal polynomials of S and T. We also consider the problem of expanding an arbitrary homogeneous symmetric polynomial, P(S) in a series of zonal polynomials. Further, these operators are used to derive identities expressing the doubly generalised binomial coefficients ( _P ^λ ), P(S) being a monomial in the power sums of the latent roots of S, in terms of the coefficients of the zonal polynomials, and from these, various results are obtained.