# §16.14(i) Appell Functions

 16.14.1 $\displaystyle x(1-x)\frac{{\partial}^{2}\mathop{{F_{1}}\/}\nolimits}{{\partial x% }^{2}}+y(1-x)\frac{{\partial}^{2}\mathop{{F_{1}}\/}\nolimits}{\partial x% \partial y}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{\partial\mathop{{F_{1}}% \/}\nolimits}{\partial x}-\beta y\frac{\partial\mathop{{F_{1}}\/}\nolimits}{% \partial y}-\alpha\beta\mathop{{F_{1}}\/}\nolimits$ $\displaystyle=0,$ $\displaystyle y(1-y)\frac{{\partial}^{2}\mathop{{F_{1}}\/}\nolimits}{{\partial y% }^{2}}+x(1-y)\frac{{\partial}^{2}\mathop{{F_{1}}\/}\nolimits}{\partial x% \partial y}+\left(\gamma-(\alpha+\beta^{\prime}+1)y\right)\frac{\partial% \mathop{{F_{1}}\/}\nolimits}{\partial y}-\beta^{\prime}x\frac{\partial\mathop{% {F_{1}}\/}\nolimits}{\partial x}-\alpha\beta^{\prime}\mathop{{F_{1}}\/}\nolimits$ $\displaystyle=0,$
 16.14.2 $\displaystyle x(1-x)\frac{{\partial}^{2}\mathop{{F_{2}}\/}\nolimits}{{\partial x% }^{2}}-xy\frac{{\partial}^{2}\mathop{{F_{2}}\/}\nolimits}{\partial x\partial y% }+\left(\gamma-(\alpha+\beta+1)x\right)\frac{\partial\mathop{{F_{2}}\/}% \nolimits}{\partial x}-\beta y\frac{\partial\mathop{{F_{2}}\/}\nolimits}{% \partial y}-\alpha\beta\mathop{{F_{2}}\/}\nolimits$ $\displaystyle=0,$ $\displaystyle y(1-y)\frac{{\partial}^{2}\mathop{{F_{2}}\/}\nolimits}{{\partial y% }^{2}}-xy\frac{{\partial}^{2}\mathop{{F_{2}}\/}\nolimits}{\partial x\partial y% }+\left(\gamma^{\prime}-(\alpha+\beta^{\prime}+1)y\right)\frac{\partial\mathop% {{F_{2}}\/}\nolimits}{\partial y}-\beta^{\prime}x\frac{\partial\mathop{{F_{2}}% \/}\nolimits}{\partial x}-\alpha\beta^{\prime}\mathop{{F_{2}}\/}\nolimits$ $\displaystyle=0,$
 16.14.3 $\displaystyle x(1-x)\frac{{\partial}^{2}\mathop{{F_{3}}\/}\nolimits}{{\partial x% }^{2}}+y\frac{{\partial}^{2}\mathop{{F_{3}}\/}\nolimits}{\partial x\partial y}% +\left(\gamma-(\alpha+\beta+1)x\right)\frac{\partial\mathop{{F_{3}}\/}% \nolimits}{\partial x}-\alpha\beta\mathop{{F_{3}}\/}\nolimits$ $\displaystyle=0,$ $\displaystyle y(1-y)\frac{{\partial}^{2}\mathop{{F_{3}}\/}\nolimits}{{\partial y% }^{2}}+x\frac{{\partial}^{2}\mathop{{F_{3}}\/}\nolimits}{\partial x\partial y}% +\left(\gamma-(\alpha^{\prime}+\beta^{\prime}+1)y\right)\frac{\partial\mathop{% {F_{3}}\/}\nolimits}{\partial y}-\alpha^{\prime}\beta^{\prime}\mathop{{F_{3}}% \/}\nolimits$ $\displaystyle=0,$
 16.14.4 $\displaystyle x(1-x)\frac{{\partial}^{2}\mathop{{F_{4}}\/}\nolimits}{{\partial x% }^{2}}-2xy\frac{{\partial}^{2}\mathop{{F_{4}}\/}\nolimits}{\partial x\partial y% }-y^{2}\frac{{\partial}^{2}\mathop{{F_{4}}\/}\nolimits}{{\partial y}^{2}}+% \left(\gamma-(\alpha+\beta+1)x\right)\frac{\partial\mathop{{F_{4}}\/}\nolimits% }{\partial x}-(\alpha+\beta+1)y\frac{\partial\mathop{{F_{4}}\/}\nolimits}{% \partial y}-\alpha\beta\mathop{{F_{4}}\/}\nolimits$ $\displaystyle=0,$ $\displaystyle y(1-y)\frac{{\partial}^{2}\mathop{{F_{4}}\/}\nolimits}{{\partial y% }^{2}}-2xy\frac{{\partial}^{2}\mathop{{F_{4}}\/}\nolimits}{\partial x\partial y% }-x^{2}\frac{{\partial}^{2}\mathop{{F_{4}}\/}\nolimits}{{\partial x}^{2}}+% \left(\gamma^{\prime}-(\alpha+\beta+1)y\right)\frac{\partial\mathop{{F_{4}}\/}% \nolimits}{\partial y}-(\alpha+\beta+1)x\frac{\partial\mathop{{F_{4}}\/}% \nolimits}{\partial x}-\alpha\beta\mathop{{F_{4}}\/}\nolimits$ $\displaystyle=0.$

# §16.14(ii) Other Functions

In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two $\mathop{{{}_{2}F_{1}}\/}\nolimits$ functions, and which satisfy pairs of linear partial differential equations of the second order. Two examples are provided by

 16.14.5 $\displaystyle G_{2}(\alpha,\alpha^{\prime};\beta,\beta^{\prime};x,y)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(% \alpha+m\right)\mathop{\Gamma\/}\nolimits\!\left(\alpha^{\prime}+n\right)% \mathop{\Gamma\/}\nolimits\!\left(\beta+n-m\right)\mathop{\Gamma\/}\nolimits\!% \left(\beta^{\prime}+m-n\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha% \right)\mathop{\Gamma\/}\nolimits\!\left(\alpha^{\prime}\right)\mathop{\Gamma% \/}\nolimits\!\left(\beta\right)\mathop{\Gamma\/}\nolimits\!\left(\beta^{% \prime}\right)}\frac{x^{m}y^{n}}{m!n!},$ $|x|<1$, $|y|<1$, 16.14.6 $\displaystyle G_{3}(\alpha,\alpha^{\prime};x,y)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(% \alpha+2n-m\right)\mathop{\Gamma\/}\nolimits\!\left(\alpha^{\prime}+2m-n\right% )}{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)\mathop{\Gamma\/}\nolimits\!% \left(\alpha^{\prime}\right)}\frac{x^{m}y^{n}}{m!n!},$ $|x|+|y|<\frac{1}{4}$.

(The region of convergence $|x|+|y|<\frac{1}{4}$ is not quite maximal.) See Erdélyi et al. (1953a, §§5.7.1–5.7.2) for further information.