# §16.1 Special Notation

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

$p,q$ nonnegative integers. nonnegative integers, unless stated otherwise. complex variable. real or complex parameters. arbitrary small positive constant. vector $(a_{1},a_{2},\dots,a_{p})$. vector $(b_{1},b_{2},\dots,b_{q})$. ${\left(a_{1}\right)_{k}}{\left(a_{2}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}$. ${\left(b_{1}\right)_{k}}{\left(b_{2}\right)_{k}}\cdots{\left(b_{q}\right)_{k}}$. $\ifrac{\mathrm{d}}{\mathrm{d}z}$. $z\ifrac{\mathrm{d}}{\mathrm{d}z}$.

The main functions treated in this chapter are the 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 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$, ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)$, ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)$, ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)$, and the 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 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 ${G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)$ for the Meijer $G$-function.