# §35.7 Gaussian Hypergeometric Function of Matrix Argument

## §35.7(i) Definition

 35.7.1 ${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},$ $-c+\frac{1}{2}(j+1)\notin\mathbb{N}$, $1\leq j\leq m$; $\|\mathbf{T}\|<1$.

### Jacobi Form

 35.7.2 $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+% \nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*% {{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1% }{2}(m+1)};\mathbf{T}\right),$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$; $\gamma,\delta,\nu\in\mathbb{C}$; $\Re\left(\gamma\right)>-1$.

## §35.7(ii) Basic Properties

### Case $m=2$

 35.7.3 ${{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).$ ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$: generalized hypergeometric function of matrix argument, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $a$: complex variable, $t_{j}$: eigenvalues of $\mathbf{T}$, $b$: complex variable, $c$: complex variable and $k$: nonnegative integer Referenced by: Erratum (V1.0.25) for Equation (35.7.3) Permalink: http://dlmf.nist.gov/35.7.E3 Encodings: TeX, pMML, png Notational change (effective with 1.0.25): Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF. See also: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

### Confluent Form

 35.7.4 $\lim_{c\to\infty}{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}-c\mathbf{T}^{-1}% \right)=\left|\mathbf{T}\right|^{b}\Psi\left(b;b-a+\tfrac{1}{2}(m+1);\mathbf{T% }\right).$

### Integral Representation

 35.7.5 ${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left% (a,c-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}-\mathbf{X}\right|}^{% c-a-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{T}\mathbf{X}\right|}^{-b}\,% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(c-a\right)>\frac{1}{2}(m-1)$, $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$.

### Transformations of Parameters

 35.7.6 ${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\left|\mathbf{I}-\mathbf{T}% \right|^{c-a-b}{{}_{2}F_{1}}\left({c-a,c-b\atop c};\mathbf{T}\right)=\left|% \mathbf{I}-\mathbf{T}\right|^{-a}{{}_{2}F_{1}}\left({a,c-b\atop c};-\mathbf{T}% (\mathbf{I}-\mathbf{T})^{-1}\right)=\left|\mathbf{I}-\mathbf{T}\right|^{-b}{{}% _{2}F_{1}}\left({c-a,b\atop c};-\mathbf{T}(\mathbf{I}-\mathbf{T})^{-1}\right).$

### Gauss Formula

 35.7.7 ${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}\right)=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(c-a-b\right)}{\Gamma_{m}\left(c-a\right)\Gamma_{m}\left% (c-b\right)}},$ $\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$.

### Reflection Formula

 35.7.8 ${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(c-a-b\right)}{\Gamma_{m}\left(c-a\right)\Gamma_{m}\left% (c-b\right)}}\*{{}_{2}F_{1}}\left({a,b\atop a+b-c+\frac{1}{2}(m+1)};\mathbf{I}% -\mathbf{T}\right),$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$; ${\frac{1}{2}}(j+1)-a\in\mathbb{N}$ for some $j=1,\ldots,m$; ${\frac{1}{2}}(j+1)-c\notin\mathbb{N}$ and $c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}$ for all $j=1,\ldots,m$. ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$: generalized hypergeometric function of matrix argument, $\in$: element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$: multivariate gamma function, $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, $\Re$: real part, $a$: complex variable, $\mathbf{I}$: $m\times m$ identity matrix, $\mathbf{T}$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable, $j$: nonnegative integer, $m$: positive integer and $\boldsymbol{{0}}$: $m\times m$ matrix of zeros Referenced by: §35.7(ii), Erratum (V1.0.26) for Equation (35.7.8) Permalink: http://dlmf.nist.gov/35.7.E8 Encodings: TeX, pMML, png Clarification (effective with 1.0.26): Originally the constraint was written as $\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$. The constraint has been replaced with $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$; ${\frac{1}{2}}(j+1)-a\in\mathbb{N}$ for some $j=1,\ldots,m$; ${\frac{1}{2}}(j+1)-c\notin\mathbb{N}$ and $c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}$ for all $j=1,\ldots,m$; for details see https://arxiv.org/abs/2002.05248. See also: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

## §35.7(iii) Partial Differential Equations

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

 35.7.9 $t_{j}(1-t_{j})\frac{{\partial}^{2}F}{{\partial t_{j}}^{2}}-\frac{1}{2}\sum_{% \begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{m}\frac{t_{k}(1-t_{k})}{t_{j}-t_{k}}\frac{\partial F}{% \partial t_{k}}+\left({c-\tfrac{1}{2}(m-1)-\left(a+b-\tfrac{1}{2}(m-3)\right)t% _{j}}+\frac{1}{2}\sum_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{m}\frac{t_{j}(1-t_{j})}{t_{j}-t_{k}}\right)\frac{% \partial F}{\partial t_{j}}=abF,$

for $j=1,\dots,m$.

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).

## §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 ${{}_{2}F_{1}}\left({\alpha a,\alpha b\atop\alpha c};\mathbf{T}\right)$

and

as $\alpha\to\infty$. 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).