§16.14 Partial Differential Equations

§16.14(i) Appell Functions

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

§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 ${{}_{2}F_{1}}$ functions, and which satisfy pairs of linear partial differential equations of the second order. Two examples are provided by

 16.14.5 $\displaystyle G_{2}(\alpha,\alpha^{\prime};\beta,\beta^{\prime};x,y)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{\Gamma\left(\alpha+m\right)\Gamma% \left(\alpha^{\prime}+n\right)\Gamma\left(\beta+n-m\right)\Gamma\left(\beta^{% \prime}+m-n\right)}{\Gamma\left(\alpha\right)\Gamma\left(\alpha^{\prime}\right% )\Gamma\left(\beta\right)\Gamma\left(\beta^{\prime}\right)}\frac{x^{m}y^{n}}{m% !n!},$ $|x|<1$, $|y|<1$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function and $!$: factorial (as in $n!$) Permalink: http://dlmf.nist.gov/16.14.E5 Encodings: TeX, pMML, png See also: Annotations for 16.14(ii), 16.14 and 16 16.14.6 $\displaystyle G_{3}(\alpha,\alpha^{\prime};x,y)$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{\Gamma\left(\alpha+2n-m\right)\Gamma% \left(\alpha^{\prime}+2m-n\right)}{\Gamma\left(\alpha\right)\Gamma\left(\alpha% ^{\prime}\right)}\frac{x^{m}y^{n}}{m!n!},$ $|x|+|y|<\frac{1}{4}$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function and $!$: factorial (as in $n!$) Permalink: http://dlmf.nist.gov/16.14.E6 Encodings: TeX, pMML, png See also: Annotations for 16.14(ii), 16.14 and 16

(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.