16 Generalized Hypergeometric Functions & Meijer G-Function Two-Variable Hypergeometric Functions 16.13 Appell Functions 16.15 Integral Representations and Integrals

§16.14 Partial Differential Equations

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Permalink:: http://dlmf.nist.gov/16.14
See also:: Annotations for Ch.16

§16.14(i) Appell Functions

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Keywords:: Appell functions, partial differential equations
Notes:: See Erdélyi et al. (1953a, §5.9).
Permalink:: http://dlmf.nist.gov/16.14.i
See also:: Annotations for §16.14 and Ch.16

16.14.1	$\displaystyle x(1-x)\frac{{\partial}^{2}{F_{1}}}{{\partial x}^{2}}+y(1-x)\frac% {\,{\partial}^{2}{F_{1}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta% +1)x\right)\frac{\partial{F_{1}}}{\partial x}-\beta y\frac{\partial{F_{1}}}{% \partial y}-\alpha\beta{F_{1}}$	$\displaystyle=0,$
16.14.1	$\displaystyle y(1-y)\frac{{\partial}^{2}{F_{1}}}{{\partial y}^{2}}+x(1-y)\frac% {\,{\partial}^{2}{F_{1}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta% ^{\prime}+1)y\right)\frac{\partial{F_{1}}}{\partial y}-\beta^{\prime}x\frac{% \partial{F_{1}}}{\partial x}-\alpha\beta^{\prime}{F_{1}}$	$\displaystyle=0,$
ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : 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	$\displaystyle x(1-x)\frac{{\partial}^{2}{F_{2}}}{{\partial x}^{2}}-xy\frac{\,{% \partial}^{2}{F_{2}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x% \right)\frac{\partial{F_{2}}}{\partial x}-\beta y\frac{\partial{F_{2}}}{% \partial y}-\alpha\beta{F_{2}}$	$\displaystyle=0,$
16.14.2	$\displaystyle y(1-y)\frac{{\partial}^{2}{F_{2}}}{{\partial y}^{2}}-xy\frac{\,{% \partial}^{2}{F_{2}}}{\,\partial x\,\partial y}+\left(\gamma^{\prime}-(\alpha+% \beta^{\prime}+1)y\right)\frac{\partial{F_{2}}}{\partial y}-\beta^{\prime}x% \frac{\partial{F_{2}}}{\partial x}-\alpha\beta^{\prime}{F_{2}}$	$\displaystyle=0,$
ⓘ Symbols: ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : 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	$\displaystyle x(1-x)\frac{{\partial}^{2}{F_{3}}}{{\partial x}^{2}}+y\frac{\,{% \partial}^{2}{F_{3}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x% \right)\frac{\partial{F_{3}}}{\partial x}-\alpha\beta{F_{3}}$	$\displaystyle=0,$
16.14.3	$\displaystyle y(1-y)\frac{{\partial}^{2}{F_{3}}}{{\partial y}^{2}}+x\frac{\,{% \partial}^{2}{F_{3}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha^{\prime}+% \beta^{\prime}+1)y\right)\frac{\partial{F_{3}}}{\partial y}-\alpha^{\prime}% \beta^{\prime}{F_{3}}$	$\displaystyle=0,$
ⓘ Symbols: ${F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$ : third Appell function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : 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	$\displaystyle x(1-x)\frac{{\partial}^{2}{F_{4}}}{{\partial x}^{2}}-2xy\frac{\,% {\partial}^{2}{F_{4}}}{\,\partial x\,\partial y}-y^{2}\frac{{\partial}^{2}{F_{% 4}}}{{\partial y}^{2}}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{\partial{F_{% 4}}}{\partial x}-(\alpha+\beta+1)y\frac{\partial{F_{4}}}{\partial y}-\alpha% \beta{F_{4}}$	$\displaystyle=0,$
16.14.4	$\displaystyle y(1-y)\frac{{\partial}^{2}{F_{4}}}{{\partial y}^{2}}-2xy\frac{\,% {\partial}^{2}{F_{4}}}{\,\partial x\,\partial y}-x^{2}\frac{{\partial}^{2}{F_{% 4}}}{{\partial x}^{2}}+\left(\gamma^{\prime}-(\alpha+\beta+1)y\right)\frac{% \partial{F_{4}}}{\partial y}-(\alpha+\beta+1)x\frac{\partial{F_{4}}}{\partial x% }-\alpha\beta{F_{4}}$	$\displaystyle=0.$
ⓘ Symbols: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : 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

§16.14(ii) Other Functions

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Permalink:: http://dlmf.nist.gov/16.14.ii
See also:: Annotations for §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	$\displaystyle G_{2}(\alpha,\alpha^{\prime};\beta,\beta^{\prime};x,y)$	$\displaystyle=\sum_{m,n=0}^{\infty}\frac{\Gamma\left(\alpha+m\right)\Gamma% \left(\alpha^{\prime}+n\right)\Gamma\left(\beta+n-m\right)\Gamma\left(\beta^{% \prime}+m-n\right)}{\Gamma\left(\alpha\right)\Gamma\left(\alpha^{\prime}\right% )\Gamma\left(\beta\right)\Gamma\left(\beta^{\prime}\right)}\frac{x^{m}y^{n}}{m% !n!},$
$\|x\|<1$ , $\|y\|<1$ ,
ⓘ 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 Ch.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 Ch.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.

16.13 Appell Functions 16.15 Integral Representations and Integrals

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§16.14 Partial Differential Equations

Contents

§16.14(i) Appell Functions

§16.14(ii) Other Functions