ALGEBRAIC ALGORITHM FOR CALCULATING COEFFICIENTS OF ZONAL POLYNOMIALS OF ORDER THREE

Many multivariate nonnull distributions and noncentral distributions involve functions which can be expanded in series of zonal polynomials, including the distribution of the covariance matrix [James (1960)], the distribution of the largest latent root and the corresponding latent vector of a Wishart matrix [Sugiyama (1966, 1967a)], and the distribution of the largest and smallest distribution of a multivariate beta distribution [Constantine (1963), Sugiyama (1967b)]. The coefficients of zonal polynomials are given by James (1964) up to degree 6 in terms of power sums and elementary symmetric functions, and by Tumura (1965) up to degree 6 in terms of monomial symmetric functions. Sugiyama (1979) has obtained zonal polynomials up to degree 200 in the case of order 2, expressed by a liner combination of monomial symmetric functions. Gupta and Richards (1979) have designed algorithms for transformation from zonal polynomial to power sums of order 3. Kowata and Wada (1992) have given a recurrence relation for the coefficients and an explicit expression of zonal polynomials of order 3 in terms of elementary symmetric functions. This expression is not so much useful for algebraic and numerical computation than that recurrence relation, because it involves a triple sum of complicated ratios of many Γ-functions with arguments among which no simple rule seems to be found. This paper discusses an algorithm based on the recurrence relation due to Kowata and Wada and gives a program package in Mathematica language.