Summary
Associated with each zonal polynomial,C k(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 ( λP ),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.
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Richards, D.S.P. Differential operators associated with zonal polynomials. I. Ann Inst Stat Math 34, 111–117 (1982). https://doi.org/10.1007/BF02481012
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DOI: https://doi.org/10.1007/BF02481012