§16.14 Partial Differential Equations

16.14.1		$x(1-x){\displaystyle \frac{{\partial }^2{F}_1}{{\partial x}^2}}+y(1-x){\displaystyle \frac{{\partial }^2{F}_1}{\partial x\partial y}}+\left(\gamma -(\alpha +\beta +1)x\right){\displaystyle \frac{\partial F_1}{\partial x}}-\beta y{\displaystyle \frac{\partial F_1}{\partial y}}-\alpha \beta F_1$	$=0,$
		$y(1-y){\displaystyle \frac{{\partial }^2{F}_1}{{\partial y}^2}}+x(1-y){\displaystyle \frac{{\partial }^2{F}_1}{\partial x\partial y}}+\left(\gamma -(\alpha +{\beta }^{\prime }+1)y\right){\displaystyle \frac{\partial F_1}{\partial y}}-{\beta }^{\prime }x{\displaystyle \frac{\partial F_1}{\partial x}}-\alpha {\beta }^{\prime }{F}_1$	$=0,$
ⓘ Symbols: $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$ Permalink: http://dlmf.nist.gov/16.14.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §16.14(i), §16.14 and Ch.16

16.14.2		$x(1-x){\displaystyle \frac{{\partial }^2{F}_2}{{\partial x}^2}}-xy{\displaystyle \frac{{\partial }^2{F}_2}{\partial x\partial y}}+\left(\gamma -(\alpha +\beta +1)x\right){\displaystyle \frac{\partial F_2}{\partial x}}-\beta y{\displaystyle \frac{\partial F_2}{\partial y}}-\alpha \beta F_2$	$=0,$
		$y(1-y){\displaystyle \frac{{\partial }^2{F}_2}{{\partial y}^2}}-xy{\displaystyle \frac{{\partial }^2{F}_2}{\partial x\partial y}}+\left({\gamma }^{\prime }-(\alpha +{\beta }^{\prime }+1)y\right){\displaystyle \frac{\partial F_2}{\partial y}}-{\beta }^{\prime }x{\displaystyle \frac{\partial F_2}{\partial x}}-\alpha {\beta }^{\prime }{F}_2$	$=0,$
ⓘ Symbols: $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$ Permalink: http://dlmf.nist.gov/16.14.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §16.14(i), §16.14 and Ch.16

16.14.3		$x(1-x){\displaystyle \frac{{\partial }^2{F}_3}{{\partial x}^2}}+y{\displaystyle \frac{{\partial }^2{F}_3}{\partial x\partial y}}+\left(\gamma -(\alpha +\beta +1)x\right){\displaystyle \frac{\partial F_3}{\partial x}}-\alpha \beta F_3$	$=0,$
		$y(1-y){\displaystyle \frac{{\partial }^2{F}_3}{{\partial y}^2}}+x{\displaystyle \frac{{\partial }^2{F}_3}{\partial x\partial y}}+\left(\gamma -({\alpha }^{\prime }+{\beta }^{\prime }+1)y\right){\displaystyle \frac{\partial F_3}{\partial y}}-{\alpha }^{\prime }{\beta }^{\prime }{F}_3$	$=0,$
ⓘ Symbols: $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$ Permalink: http://dlmf.nist.gov/16.14.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §16.14(i), §16.14 and Ch.16

16.14.4		$x(1-x){\displaystyle \frac{{\partial }^2{F}_4}{{\partial x}^2}}-2xy{\displaystyle \frac{{\partial }^2{F}_4}{\partial x\partial y}}-y^2{\displaystyle \frac{{\partial }^2{F}_4}{{\partial y}^2}}+\left(\gamma -(\alpha +\beta +1)x\right){\displaystyle \frac{\partial F_4}{\partial x}}-(\alpha +\beta +1)y{\displaystyle \frac{\partial F_4}{\partial y}}-\alpha \beta F_4$	$=0,$
		$y(1-y){\displaystyle \frac{{\partial }^2{F}_4}{{\partial y}^2}}-2xy{\displaystyle \frac{{\partial }^2{F}_4}{\partial x\partial y}}-x^2{\displaystyle \frac{{\partial }^2{F}_4}{{\partial x}^2}}+\left({\gamma }^{\prime }-(\alpha +\beta +1)y\right){\displaystyle \frac{\partial F_4}{\partial y}}-(\alpha +\beta +1)x{\displaystyle \frac{\partial F_4}{\partial x}}-\alpha \beta F_4$	$=0.$
ⓘ Symbols: $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$ Permalink: http://dlmf.nist.gov/16.14.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §16.14(i), §16.14 and Ch.16

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		$G_2(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };x,y)$	$={\displaystyle \sum _{m,n=0}^{\mathrm{\infty }}}{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +m\right)\mathrm{\Gamma }\left({\alpha }^{\prime }+n\right)\mathrm{\Gamma }\left(\beta +n-m\right)\mathrm{\Gamma }\left({\beta }^{\prime }+m-n\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left({\alpha }^{\prime }\right)\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)}}{\displaystyle \frac{x^m{y}^n}{m!n!}},$
, ,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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 Ch.16
16.14.6		$G_3(\alpha ,{\alpha }^{\prime };x,y)$	$={\displaystyle \sum _{m,n=0}^{\mathrm{\infty }}}{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +2n-m\right)\mathrm{\Gamma }\left({\alpha }^{\prime }+2m-n\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left({\alpha }^{\prime }\right)}}{\displaystyle \frac{x^m{y}^n}{m!n!}},$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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 Ch.16

(The region of convergence is not quite maximal.) See Erdélyi et al. (1953a, §§5.7.1–5.7.2) for further information.