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35 Functions of Matrix ArgumentProperties

§35.7 Gaussian Hypergeometric Function of Matrix Argument

  1. §35.7(i) Definition
  2. §35.7(ii) Basic Properties
  3. §35.7(iii) Partial Differential Equations
  4. §35.7(iv) Asymptotic Approximations

§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(ca)k(b)k(cb)kk!(c)2k(c12)k(t1t2)k×F12(a+k,b+kc+2k;t1+t2t1t2).

Confluent Form

Integral Representation

Transformations of Parameters

35.7.6 F12(a,bc;𝐓)=|𝐈𝐓|cabF12(ca,cbc;𝐓)=|𝐈𝐓|aF12(a,cbc;𝐓(𝐈𝐓)1)=|𝐈𝐓|bF12(ca,bc;𝐓(𝐈𝐓)1).

Gauss Formula

Reflection Formula

35.7.8 F12(a,bc;𝐓)=Γm(c)Γm(cab)Γm(ca)Γm(cb)F12(a,ba+bc+12(m+1);𝐈𝐓),
𝟎<𝐓<𝐈; 12(j+1)a for some j=1,,m; 12(j+1)c and cab12(mj) for all j=1,,m.

§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(1tj)2Ftj212k=1kjmtk(1tk)tjtkFtk+(c12(m1)(a+b12(m3))tj+12k=1kjmtj(1tj)tjtk)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)


35.7.11 F12(a,bc;𝐈α1𝐓)

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).