ALGEBRAIC ALGORITHM FOR CALCULATING COEFFICIENTS OF ZONAL POLYNOMIALS OF ORDER THREE

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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.
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ALGEBRAIC ALGORITHM FOR CALCULATING COEFFICIENTS OF ZONAL POLYNOMIALS OF ORDER THREE

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