An Asymptotic Expansion for the Distribution of the Eigenvalues of a 3 by 3 Wishart Matrix

Abstract

A parametrization of the rotation group $O^+(p)$ of $p$ by $p$ orthogonal matrices with determinant $+1$ in terms of their skew symmetric parts is used to derive, for $p = 3$, an explicit expansion for $_0F_0^{(p)}(Z, \Omega)$, a hypergeometric function of two matrix arguments appearing in the distribution of the eigenvalues of a $p$ by $p$ Wishart matrix. On the basis of a numerically derived simplification of the low order terms of this series, an asymptotic expansion of $_0F_0^{(3)}$ in terms of products of ordinary confluent hypergeometric series is conjectured. Limited numerical exploration indicates the new series to be several orders of magnitude more accurate than the series from which it was derived.