System of Partial Differential Equations for the Hypergeometric Function 1F1 of a Matrix Argument on Diagonal Regions

Abstract

The hypergeometric function 1F1 of a matrix argument Y is a symmetric entire function in the eigenvalues y1,...,ym of Y. It appears in the distribution function of the largest eigenvalue of a Wishart matrix and its numerical evaluation is important in multivariate distribution theory. Hashiguchi et al. (J. Multivariate Analysis, 2013) proposed an efficient algorithm for evaluating the matrix 1F1 by the holonomic gradient method (HGM). The algorithm is based on the system of partial differential equations (PDEs) satisfied by the matrix 1F1 given by Muirhead (Ann. Math. Statist., 1970) and it cannot be applied to the diagonal cases, i.e. the cases where several yi's are equal because the system of PDEs has singularities on the diagonal region. Hashiguchi et al. derived an ordinary differential equation (ODE) satisfied by 1F1(y,y) in the bivariate case from some relations which are obtained by applying l'opital rule to the system of PDEs for 1F1(y1,y2). In this paper we generalize this approach for computing systems of PDEs satisfied by the matrix 1F1 for various diagonalization patterns. We show that the existence of a system of PDEs for a diagonalized 1F1 is reduced to the non-singularity of the matrices systematically derived from the diagonalization pattern. By checking the non-singularity numerically, we show that there exists a system of PDEs for a diagonalized 1F1 if the size of each diagonal block ≤ 36. We have computed an ODE for 1F1(y,...,y) up to m=22. We made a test implementation of HGM for diagonal cases and we show some numerical results.