# §35.7 Gaussian Hypergeometric Function of Matrix Argument

35.7.1, ; .

35.7.4

## §35.7(iii) Partial Differential Equations

Let (a) be orthogonally invariant, so that is a symmetric function of , the eigenvalues of the matrix argument ; (b) be analytic in in a neighborhood of ; (c) satisfy . Subject to the conditions (a)–(c), the function is the unique solution of each partial differential equation

for .

Systems of partial differential equations for the (defined in §35.8) and functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9).

## §35.7(iv) Asymptotic Approximations

Butler and Wood (2002) applies Laplace’s method (§2.3(iii)) to (35.7.5) to derive uniform asymptotic approximations for the functions

and

as . These approximations are in terms of elementary functions.

For other asymptotic approximations for Gaussian hypergeometric functions of matrix argument, see Herz (1955), Muirhead (1982, pp. 264–281, 290, 472, 563), and Butler and Wood (2002).