16Generalized Hypergeometric Functions & MeijerG-Function Notation 16Generalized Hypergeometric Functions & MeijerG-Function 16.2Definition and Analytic Properties

§16.1Special Notation

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Permalink:: http://dlmf.nist.gov/16.1
See also:: Annotations forCh.16

(For other notation seeNotation 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 $(b_{1},b_{2},\dots,b_{q})$ .
${\left(\mathbf{a}\right)_{k}}$	${\left(a_{1}\right)_{k}}{\left(a_{2}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}$ .
${\left(\mathbf{b}\right)_{k}}$	${\left(b_{1}\right)_{k}}{\left(b_{2}\right)_{k}}\cdots{\left(b_{q}\right)_{k}}$ .
$D$	$\ifrac{\mathrm{d}}{\mathrm{d}z}$ .
$\vartheta$	$z\ifrac{\mathrm{d}}{\mathrm{d}z}$ .

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

16Generalized Hypergeometric Functions & MeijerG-Function 16.2Definition and Analytic Properties

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§16.1Special Notation