# §35.7 Gaussian Hypergeometric Function of Matrix Argument

## §35.7(i) Definition

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

### Jacobi Form

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

## §35.7(ii) Basic Properties

### Case $m=2$

 35.7.3 $\mathop{{{}_{2}F_{1}}\/}\nolimits\!\left({a,b\atop c};\left(\begin{matrix}t_{1% }&0\\ 0&t_{2}\end{matrix}\right)\right)=\sum_{k=0}^{\infty}\frac{\left(a\right)_{k}% \left(c-a\right)_{k}\left(b\right)_{k}\left(c-b\right)_{k}}{k!\,\left(c\right)% _{2k}\left(c-\tfrac{1}{2}\right)_{k}}\*(t_{1}t_{2})^{k}\mathop{{{}_{2}F_{1}}\/% }\nolimits\!\left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).$

### Confluent Form

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

### Integral Representation

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

### Transformations of Parameters

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

### Gauss Formula

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

### Reflection Formula

 35.7.8 ${\mathop{{{}_{2}F_{1}}\/}\nolimits\!\left({a,b\atop c};\mathbf{T}\right)=\frac% {\mathop{\Gamma_{m}\/}\nolimits\!\left(c\right)\mathop{\Gamma_{m}\/}\nolimits% \!\left(c-a-b\right)}{\mathop{\Gamma_{m}\/}\nolimits\!\left(c-a\right)\mathop{% \Gamma_{m}\/}\nolimits\!\left(c-b\right)}}\*\mathop{{{}_{2}F_{1}}\/}\nolimits% \!\left({a,b\atop a+b-c+\frac{1}{2}(m+1)};\mathbf{I}-\mathbf{T}\right),$ $\realpart{(c)},\realpart{(c-a-b)}>\frac{1}{2}(m-1)$.

## §35.7(iii) Partial Differential Equations

Let $f:{\boldsymbol{\Omega}}\to\Complex$ (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 $f(\mathbf{T})=\mathop{{{}_{2}F_{1}}\/}\nolimits\!\left(a,b;c;\mathbf{T}\right)$ 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 $\mathop{{{}_{0}F_{1}}\/}\nolimits$ (defined in §35.8) and $\mathop{{{}_{1}F_{1}}\/}\nolimits$ 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 $\alpha\to\infty$. 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).