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 p F q of matrix arqument. Muirhead [6] and Constantine and Muirhead [2] gave partial differential equations for the functions of 2 F 1 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 3 F 2 (a 1 a 2 , a 3 ; b 1 , b 2 , R) and 3 F 2 (a 1 , a 2 , a 3 ; b 1 , b 2 ; R, S). Differential equations for 2 F 2 are also obtained.