Differential operators associated with zonal polynomials. I

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Abstract

Associated with each zonal polynomial, C k(S), of a symmetric matrix S, we define a differential operator ∂k, having the basic property that ∂kCλδ, 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.

Original languageEnglish (US)
Pages (from-to)111-117
Number of pages7
JournalAnnals of the Institute of Statistical Mathematics
Volume34
Issue number1
DOIs
StatePublished - Dec 1 1982

All Science Journal Classification (ASJC) codes

  • Statistics and Probability

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