Zonal Polynomials

Abstract

The first section of this chapter is devoted to some properties of Wishart distribution. Section 4.2 gives some relations between certain symmetric functions and elementary symmetric functions. Zonal polynomials of a matrix argument are constructed in Section 4.3. Section 4.4 deals with the Laplace transform of a tonal polynomial and hypergeometric functions of matrix argument. Some weighted sums of tonal polynomials are also given in this section. Section 4.5 introduces two types of binomial coefficients which are extensions of a usual binomial coefficient. Section 4.6 constructs some special functions, Laguerre polynomials, Hermite polynomials and P-polynomials, and gives generating functions of these which are used for the derivation of the probability density function of a generalized quadratic form.