# §16.13 Appell Functions

The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two $\mathop{{{}_{2}F_{1}}\/}\nolimits$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1):

 16.13.1 $\displaystyle\mathop{{F_{1}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};% \gamma;x,y\right)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{\left(\alpha\right)_{m+n}\left(\beta% \right)_{m}\left(\beta^{\prime}\right)_{n}}{\left(\gamma\right)_{m+n}m!n!}x^{m% }y^{n},$ $\max\left(|x|,|y|\right)<1$, Defines: $\mathop{{F_{1}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$: Appell function Symbols: $\left(a\right)_{n}$: Pochhammer’s symbol, $!$: $n!$: factorial and $(a,b)$: open interval Permalink: http://dlmf.nist.gov/16.13.E1 Encodings: TeX, pMML, png 16.13.2 $\displaystyle\mathop{{F_{2}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};% \gamma,\gamma^{\prime};x,y\right)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{\left(\alpha\right)_{m+n}\left(\beta% \right)_{m}\left(\beta^{\prime}\right)_{n}}{\left(\gamma\right)_{m}\left(% \gamma^{\prime}\right)_{n}m!n!}x^{m}y^{n},$ $|x|+|y|<1$, Defines: $\mathop{{F_{2}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{% \prime};x,y\right)$: Appell function Symbols: $\left(a\right)_{n}$: Pochhammer’s symbol and $!$: $n!$: factorial Permalink: http://dlmf.nist.gov/16.13.E2 Encodings: TeX, pMML, png 16.13.3 $\displaystyle\mathop{{F_{3}}\/}\nolimits\!\left(\alpha,\alpha^{\prime};\beta,% \beta^{\prime};\gamma;x,y\right)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{\left(\alpha\right)_{m}\left(\alpha^{% \prime}\right)_{n}\left(\beta\right)_{m}\left(\beta^{\prime}\right)_{n}}{\left% (\gamma\right)_{m+n}m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$, Defines: $\mathop{{F_{3}}\/}\nolimits\!\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime}% ;\gamma;x,y\right)$: Appell function Symbols: $\left(a\right)_{n}$: Pochhammer’s symbol, $!$: $n!$: factorial and $(a,b)$: open interval Permalink: http://dlmf.nist.gov/16.13.E3 Encodings: TeX, pMML, png 16.13.4 $\displaystyle\mathop{{F_{4}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{% \prime};x,y\right)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{\left(\alpha\right)_{m+n}\left(\beta% \right)_{m+n}}{\left(\gamma\right)_{m}\left(\gamma^{\prime}\right)_{n}m!n!}x^{% m}y^{n},$ $\sqrt{|x|}+\sqrt{|y|}<1$. Defines: $\mathop{{F_{4}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{\prime};x,y\right)$: Appell function Symbols: $\left(a\right)_{n}$: Pochhammer’s symbol and $!$: $n!$: factorial Permalink: http://dlmf.nist.gov/16.13.E4 Encodings: TeX, pMML, png

Here and elsewhere it is assumed that neither of the bottom parameters $\gamma$ and $\gamma^{\prime}$ is a nonpositive integer.

For large parameter asymptotics see López et al. (2013).