§16.1 Special Notation

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(For other notation see Notation for the Special Functions.)

$p,q$	nonnegative integers.
$k,n$	nonnegative integers, unless
	stated otherwise.
$z$	complex variable.
$\rselection{a_{1},a_{2},\dots,a_{p}\\ b_{1},b_{2},\dots,b_{q}}$	real or complex parameters.
$\delta$	arbitrary small positive constant.
$\mathbf{a}$	vector $(a_{1},a_{2},\dots,a_{p})$ .
$\mathbf{b}$	vector $(b_{1},b_{2},\dots,b_{q})$ .
$\left(\mathbf{a}\right)_{{k}}$	$\left(a_{1}\right)_{{k}}\left(a_{2}\right)_{{k}}\cdots\left(a_{p}\right)_{{k}}$ .
$\left(\mathbf{b}\right)_{{k}}$	$\left(b_{1}\right)_{{k}}\left(b_{2}\right)_{{k}}\cdots\left(b_{q}\right)_{{k}}$ .
$D$	$\ifrac{d}{dz}$ .
$\vartheta$	$z\ifrac{d}{dz}$ .

The main functions treated in this chapter are the generalized hypergeometric function $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ , the Appell (two-variable hypergeometric) functions $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$ , $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,\gamma^{{\prime}};x,y\right)$ , $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\alpha^{{\prime}};\beta,\beta^{{\prime}};\gamma;x,y\right)$ , $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y\right)$ , and the Meijer $G$ -function $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$ . Alternative notations are $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ , $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)$ , and $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ for the generalized hypergeometric function, $F_{1}(\alpha,\beta,\beta^{{\prime}};\gamma;x,y)$ , $F_{2}(\alpha,\beta,\beta^{{\prime}};\gamma,\gamma^{{\prime}};x,y)$ , $F_{3}(\alpha,\alpha^{{\prime}},\beta,\beta^{{\prime}};\gamma;x,y)$ , $F_{4}(\alpha,\beta;\gamma,\gamma^{{\prime}};x,y)$ , for the Appell functions, and $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;\mathbf{a};\mathbf{b}\right)$ for the Meijer $G$ -function.