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 ∂kCλδkλ, 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 language | English (US) |
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Pages (from-to) | 111-117 |
Number of pages | 7 |
Journal | Annals of the Institute of Statistical Mathematics |
Volume | 34 |
Issue number | 1 |
DOIs | |
State | Published - Dec 1982 |
All Science Journal Classification (ASJC) codes
- Statistics and Probability