# §35.8 Generalized Hypergeometric Functions of Matrix Argument

## §35.8(i) Definition

Let $p$ and $q$ be nonnegative integers; $a_{1},\dots,a_{p}\in\mathbb{C}$; $b_{1},\dots,b_{q}\in\mathbb{C}$; $-b_{j}+\tfrac{1}{2}(k+1)\notin\mathbb{N}$, $1\leq j\leq q$, $1\leq k\leq m$. The generalized hypergeometric function ${{}_{p}F_{q}}$ with matrix argument $\mathbf{T}\in\boldsymbol{\mathcal{S}}$, numerator parameters $a_{1},\dots,a_{p}$, and denominator parameters $b_{1},\dots,b_{q}$ is

 35.8.1 ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right% )=\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{|\kappa|=k}\frac{{\left[a_{1}\right]_{% \kappa}}\cdots{\left[a_{p}\right]_{\kappa}}}{{\left[b_{1}\right]_{\kappa}}% \cdots{\left[b_{q}\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right).$

### Convergence Properties

If $-a_{j}+\tfrac{1}{2}(k+1)\in\mathbb{N}$ for some $j,k$ satisfying $1\leq j\leq p$, $1\leq k\leq m$, then the series expansion (35.8.1) terminates.

If $p\leq q$, then (35.8.1) converges for all $\mathbf{T}$.

If $p=q+1$, then (35.8.1) converges absolutely for $||\mathbf{T}||<1$ and diverges for $||\mathbf{T}||>1$.

If $p>q+1$, then (35.8.1) diverges unless it terminates.

## §35.8(ii) Relations to Other Functions

 35.8.2 ${{}_{0}F_{0}}\left({-\atop-};\mathbf{T}\right)=\mathrm{etr}\left(\mathbf{T}% \right),$ $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
 35.8.3 ${{}_{2}F_{1}}\left({a,b\atop b};\mathbf{T}\right)={{}_{1}F_{0}}\left({a\atop-}% ;\mathbf{T}\right)=|\mathbf{I}-\mathbf{T}|^{-a},$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$.
 35.8.4 $A_{\nu}\left(\mathbf{T}\right)=\dfrac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(m+1)% \right)}{{}_{0}F_{1}}\left({-\atop\nu+\frac{1}{2}(m+1)};-\mathbf{T}\right),$ $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.

## §35.8(iii) ${{}_{3}F_{2}}$ Case

### Kummer Transformation

Let $c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}$. Then

 35.8.5 ${{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{2}\right)\Gamma_{m}\left(c\right)}{\Gamma_{m}\left(b_% {2}-a_{3}\right)\Gamma_{m}\left(c+a_{3}\right)}\*{{}_{3}F_{2}}\left({b_{1}-a_{% 1},b_{1}-a_{2},a_{3}\atop b_{1},c+a_{3}};\mathbf{I}\right),$ $\Re(b_{2}),\Re(c)>\frac{1}{2}(m-1)$.

### Pfaff–Saalschutz Formula

Let $a_{1}+a_{2}+a_{3}+\frac{1}{2}(m+1)=b_{1}+b_{2}$; one of the $a_{j}$ be a negative integer; $\Re(b_{1}-a_{1})$, $\Re(b_{1}-a_{2})$, $\Re(b_{1}-a_{3})$, $\Re(b_{1}-a_{1}-a_{2}-a_{3})>\frac{1}{2}(m-1)$. Then

 35.8.6 ${{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{1}-a_{1}\right)\Gamma_{m}\left(b_{1}-a_{2}\right)}{% \Gamma_{m}\left(b_{1}\right)\Gamma_{m}\left(b_{1}-a_{1}-a_{2}\right)}\*\frac{% \Gamma_{m}\left(b_{1}-a_{3}\right)\Gamma_{m}\left(b_{1}-a_{1}-a_{2}-a_{3}% \right)}{\Gamma_{m}\left(b_{1}-a_{1}-a_{3}\right)\Gamma_{m}\left(b_{1}-a_{2}-a% _{3}\right)}.$

### Thomae Transformation

Again, let $c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}$. Then

 35.8.7 ${{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{1}\right)\Gamma_{m}\left(b_{2}\right)\Gamma\left(c% \right)}{\Gamma_{m}\left(a_{1}\right)\Gamma_{m}\left(c+a_{2}\right)\Gamma\left% (c+a_{3}\right)}\*{{}_{3}F_{2}}\left({b_{1}-a_{1},b_{2}-a_{2},c\atop c+a_{2},c% +a_{3}};\mathbf{I}\right),$ $\Re(b_{1})$, $\Re(b_{2})$, $\Re(c)>\frac{1}{2}(m-1)$.

## §35.8(iv) General Properties

### Value at $\mathbf{T}=\boldsymbol{{0}}$

 35.8.8 ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\boldsymbol{{0}}% \right)=1.$

### Confluence

 35.8.9 $\lim_{\gamma\to\infty}{{}_{p+1}F_{q}}\left({a_{1},\dots,a_{p},\gamma\atop b_{1% },\dots,b_{q}};\gamma^{-1}\mathbf{T}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_% {p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right),$
 35.8.10 $\lim_{\gamma\to\infty}{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots% ,b_{q},\gamma};\gamma\mathbf{T}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}% \atop b_{1},\dots,b_{q}};\mathbf{T}\right).$

### Invariance

 35.8.11 ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{H}% \mathbf{T}\mathbf{H}^{-1}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_% {1},\dots,b_{q}};\mathbf{T}\right),$ $\mathbf{H}\in\mathbf{O}(m)$.

### Laplace Transform

 35.8.12 ${\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)|% \mathbf{X}|^{\gamma-\frac{1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}% \atop b_{1},\dots,b_{q}};-\mathbf{X}\right)\mathrm{d}\mathbf{X}}=\Gamma_{m}% \left(\gamma\right)|\mathbf{T}|^{-\gamma}{{}_{p+1}F_{q}}\left({a_{1},\dots,a_{% p},\gamma\atop b_{1},\dots,b_{q}};-\mathbf{T}^{-1}\right),$ $\Re(\gamma)>\frac{1}{2}(m-1)$.

### Euler Integral

 35.8.13 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}|\mathbf{X}|^{a_{1}-\frac{% 1}{2}(m+1)}{|\mathbf{I}-\mathbf{X}|}^{b_{1}-a_{1}-\frac{1}{2}(m+1)}\*{{}_{p}F_% {q}}\left({a_{2},\dots,a_{p+1}\atop b_{2},\dots,b_{q+1}};\mathbf{T}\mathbf{X}% \right)\mathrm{d}\mathbf{X}=\frac{1}{\mathrm{B}_{m}\left(b_{1}-a_{1},a_{1}% \right)}{{}_{p+1}F_{q+1}}\left({a_{1},\dots,a_{p+1}\atop b_{1},\dots,b_{q+1}};% \mathbf{T}\right),$ $\Re(b_{1}-a_{1}),\Re(a_{1})>\frac{1}{2}(m-1)$.

## §35.8(v) Mellin–Barnes Integrals

Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions ${{}_{p}F_{q}}$ and ${{}_{p+1}F_{p}}$ of matrix argument. A similar result for the ${{}_{0}F_{1}}$ function of matrix argument is given in Faraut and Korányi (1994, p. 346). These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case $m=1$.