## 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 ∂_{k}C_{λ}δ_{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.

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 1 1982 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability