A holonomic systems approach to special functions identities
Journal of Computational and Applied Mathematics
Advances in Applied Mathematics
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Properties and applications of Fisher distribution on the rotation group
Journal of Multivariate Analysis
International Journal of Knowledge Engineering and Soft Data Paradigms
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We apply the holonomic gradient method introduced by Nakayama et al. (2011) [23] to the evaluation of the exact distribution function of the largest root of a Wishart matrix, which involves a hypergeometric function "1F"1 of a matrix argument. Numerical evaluation of the hypergeometric function has been one of the longstanding problems in multivariate distribution theory. The holonomic gradient method offers a totally new approach, which is complementary to the infinite series expansion around the origin in terms of zonal polynomials. It allows us to move away from the origin by the use of partial differential equations satisfied by the hypergeometric function. From the numerical viewpoint we show that the method works well up to dimension 10. From the theoretical viewpoint the method offers many challenging problems both to statistics and D-module theory.