§35.7 Gaussian Hypergeometric Function of Matrix Argument

35.7.1		$${}_{2}F_1(\genfrac{}{}{0pt}{}{a,b}{c};\mathbf{T})=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }{\left[b\right]}_{\kappa }}{{\left[c\right]}_{\kappa }}{Z}_{\kappa }\left(\mathbf{T}\right),$$
$-c+\frac{1}{2}(j+1)\notin \mathbb{N}$, $1\le j\le m$; .
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\mathbf{T})$: generalized hypergeometric function of matrix argument, $!$: factorial (as in $n!$), $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(\mathbf{T}\right)$: zonal polynomial, $a$: complex variable, $\mathbf{T}$: real symmetric $m\times m$ matrix, $b$: complex variable, $\parallel \mathbf{T}\parallel$: spectral norm of $\mathbf{T}$, $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.7.E1 Encodings: TeX, pMML, png See also: Annotations for §35.7(i), §35.7 and Ch.35

Let $f:\mathbf{\Omega }\to ℂ$ (a) be orthogonally invariant, so that $f(\mathbf{T})$ is a symmetric function of $t_1,\mathrm{\dots },t_m$, the eigenvalues of the matrix argument $\mathbf{T}\in \mathbf{\Omega }$; (b) be analytic in $t_1,\mathrm{\dots },t_m$ in a neighborhood of $\mathbf{T}=\mathbf{0}$; (c) satisfy $f(\mathbf{0})=1$. Subject to the conditions (a)–(c), the function $f(\mathbf{T})={}_{2}F_1(a,b;c;\mathbf{T})$ is the unique solution of each partial differential equation

35.7.9		$$\begin{array}{l}\begin{array}{l}{t}_j(1-t_j)\frac{{\partial }^{2}F}{{\partial t_j}^2}-\frac{1}{2}\sum _{\begin{array}{c}\hfill k=1\hfill \\ \hfill k\ne j\hfill \end{array}}^m\frac{t_k(1-t_k)}{t_j-t_k}\frac{\partial F}{\partial t_k}\\ +\left(c-\frac{1}{2}(m-1)-\left(a+b-\frac{1}{2}(m-3)\right)t_j+\frac{1}{2}\sum _{\begin{array}{c}\hfill k=1\hfill \\ \hfill k\ne j\hfill \end{array}}^m\frac{t_j(1-t_j)}{t_j-t_k}\right)\frac{\partial F}{\partial t_j}\end{array}\\ =abF,\end{array}$$
ⓘ Symbols: $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $a$: complex variable, $t_j$: eigenvalues of $\mathbf{T}$, $b$: complex variable, $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer and $m$: positive integer Referenced by: §35.7(iii) Permalink: http://dlmf.nist.gov/35.7.E9 Encodings: TeX, pMML, png See also: Annotations for §35.7(iii), §35.7 and Ch.35

for $j=1,\mathrm{\dots },m$.

Systems of partial differential equations for the ${}_{0}F_1$ (defined in §35.8) and ${}_{1}F_1$ functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9).

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		$${}_{2}F_1(\genfrac{}{}{0pt}{}{\alpha a,\alpha b}{\alpha c};\mathbf{T})$$
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\mathbf{T})$: generalized hypergeometric function of matrix argument, $a$: complex variable, $\mathbf{T}$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter Permalink: http://dlmf.nist.gov/35.7.E10 Encodings: TeX, pMML, png See also: Annotations for §35.7(iv), §35.7 and Ch.35

and

35.7.11		$${}_{2}F_1(\genfrac{}{}{0pt}{}{a,b}{c};\mathbf{I}-{\alpha }^{-1}\mathbf{T})$$
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};\mathbf{T})$: generalized hypergeometric function of matrix argument, $a$: complex variable, $\mathbf{I}$: $m\times m$ identity matrix, $\mathbf{T}$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter Permalink: http://dlmf.nist.gov/35.7.E11 Encodings: TeX, pMML, png See also: Annotations for §35.7(iv), §35.7 and Ch.35

as $\alpha \to \mathrm{\infty }$. 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).