Differential operators associated with zonal polynomials. I

Abstract

Associated with each zonal polynomial,Ck(S), of a symmetric matrixS, we define a differential operator ∂k, having the basic property that ∂kCλδkλ, δ being Kronecker's delta, whenever κ and λ are partitions of the non-negative integerk. 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 matricesS andT, in terms of the zonal polynomials ofS andT. 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 (Pλ),P(S) being a monomial in the power sums of the latent roots ofS, in terms of the coefficients of the zonal polynomials, and from these, various results are obtained.