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35 Functions of Matrix ArgumentProperties

§35.7 Gaussian Hypergeometric Function of Matrix Argument

Contents

§35.7(i) Definition

Jacobi Form

35.7.2 Pν(γ,δ)(T)=Γm(γ+ν+12(m+1))Γm(γ+12(m+1))F12(-ν,γ+δ+ν+12(m+1)γ+12(m+1);T),
0<T<I; γ,δ,ν; (γ)>-1.

§35.7(ii) Basic Properties

Case m=2

35.7.3 F12(a,bc;(t100t2))=k=0(a)k(c-a)k(b)k(c-b)kk!(c)2k(c-12)k(t1t2)kF12(a+k,b+kc+2k;t1+t2-t1t2).

Confluent Form

Integral Representation

35.7.5 F12(a,bc;T)=1Bm(a,c-a)0<X<I|X|a-12(m+1)|I-X|c-a-12(m+1)|I-TX|-bdX,
(a),(c-a)>12(m-1), 0<T<I.

Transformations of Parameters

35.7.6 F12(a,bc;T)=|I-T|c-a-bF12(c-a,c-bc;T)=|I-T|-aF12(a,c-bc;-T(I-T)-1)=|I-T|-bF12(c-a,bc;-T(I-T)-1).

Gauss Formula

35.7.7 F12(a,bc;I)=Γm(c)Γm(c-a-b)Γm(c-a)Γm(c-b),
(c),(c-a-b)>12(m-1).

Reflection Formula

35.7.8 F12(a,bc;T)=Γm(c)Γm(c-a-b)Γm(c-a)Γm(c-b)F12(a,ba+b-c+12(m+1);I-T),
(c),(c-a-b)>12(m-1).

§35.7(iii) Partial Differential Equations

Let f:Ω (a) be orthogonally invariant, so that f(T) is a symmetric function of t1,,tm, the eigenvalues of the matrix argument TΩ; (b) be analytic in t1,,tm in a neighborhood of T=0; (c) satisfy f(0)=1. Subject to the conditions (a)–(c), the function f(T)=F12(a,b;c;T) is the unique solution of each partial differential equation

35.7.9 tj(1-tj)2Ftj2-12k=1kjmtk(1-tk)tj-tkFtk+(c-12(m-1)-(a+b-12(m-3))tj+12k=1kjmtj(1-tj)tj-tk)Ftj=abF,

for j=1,,m.

Systems of partial differential equations for the F10 (defined in §35.8) and F11 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

35.7.10 F12(αa,αbαc;T)

and

35.7.11 F12(a,bc;I-α-1T)

as α. 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).