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

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

## §35.8(iii) $\mathop{{{}_{3}F_{2}}\/}\nolimits$ Case

### Kummer Transformation

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

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

### 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 $\mathop{{{}_{3}F_{2}}\/}\nolimits\!\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};% \mathbf{I}\right)=\frac{\mathop{\Gamma_{m}\/}\nolimits\!\left(b_{1}-a_{1}% \right)\mathop{\Gamma_{m}\/}\nolimits\!\left(b_{1}-a_{2}\right)}{\mathop{% \Gamma_{m}\/}\nolimits\!\left(b_{1}\right)\mathop{\Gamma_{m}\/}\nolimits\!% \left(b_{1}-a_{1}-a_{2}\right)}\*\frac{\mathop{\Gamma_{m}\/}\nolimits\!\left(b% _{1}-a_{3}\right)\mathop{\Gamma_{m}\/}\nolimits\!\left(b_{1}-a_{1}-a_{2}-a_{3}% \right)}{\mathop{\Gamma_{m}\/}\nolimits\!\left(b_{1}-a_{1}-a_{3}\right)\mathop% {\Gamma_{m}\/}\nolimits\!\left(b_{1}-a_{2}-a_{3}\right)}.$

### Thomae Transformation

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

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

## §35.8(iv) General Properties

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

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

### Confluence

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

### Invariance

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

### Laplace Transform

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

### Euler Integral

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

## §35.8(v) Mellin–Barnes Integrals

Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions $\mathop{{{}_{p}F_{q}}\/}\nolimits$ and $\mathop{{{}_{p+1}F_{p}}\/}\nolimits$ of matrix argument. A similar result for the $\mathop{{{}_{0}F_{1}}\/}\nolimits$ 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 $m=1$.