Partial differential equations for hypergeometric functions 3F2 of matrix argument

Many multivariate non-null distributions and moment formulas can be expressed in terms of hypergeometric functions _pF_q of matrix arqument. Muirhead [6] and Constantine and Muirhead [2] gave partial differential equations for the functions of ₂F₁ of one argument matrix and two argument matrices, respectively. Such differential equations have been used to obtain asymptotic expansions of the functions (Muirhead [7], [8], [9], Sugiura [10]). The purpose of this paper is to derive partial differential equations for the functions ₃F₂ (a₁ a₂, a₃; b₁, b₂, R) and ₃F₂ (a₁, a₂, a₃; b₁, b₂; R, S). Differential equations for ₂F₂ are also obtained.