# §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 ${{}_{2}F_{1}}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1):

 16.13.1 $\displaystyle{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(% \beta\right)_{m}}{\left(\beta^{\prime}\right)_{n}}}{{\left(\gamma\right)_{m+n}% }m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$, 16.13.2 $\displaystyle{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x% ,y\right)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(% \beta\right)_{m}}{\left(\beta^{\prime}\right)_{n}}}{{\left(\gamma\right)_{m}}{% \left(\gamma^{\prime}\right)_{n}}m!n!}x^{m}y^{n},$ $|x|+|y|<1$, 16.13.3 $\displaystyle{F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x% ,y\right)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m}}{\left(% \alpha^{\prime}\right)_{n}}{\left(\beta\right)_{m}}{\left(\beta^{\prime}\right% )_{n}}}{{\left(\gamma\right)_{m+n}}m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$, 16.13.4 $\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(% \beta\right)_{m+n}}}{{\left(\gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n% }}m!n!}x^{m}y^{n},$ $\sqrt{|x|}+\sqrt{|y|}<1$.

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. (2013a, b), and Ferreira et al. (2013a, b).