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
§16.14(ii) Other Functions

§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

$x(1-x)\frac{{\partial}^{2}{F_{1}}}{{\partial x}^{2}}+y(1-x)\frac{\,{\partial}^% {2}{F_{1}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)% \frac{\partial{F_{1}}}{\partial x}-\beta y\frac{\partial{F_{1}}}{\partial y}-% \alpha\beta{F_{1}}=0,$

$y(1-y)\frac{{\partial}^{2}{F_{1}}}{{\partial y}^{2}}+x(1-y)\frac{\,{\partial}^% {2}{F_{1}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta^{\prime}+1)y% \right)\frac{\partial{F_{1}}}{\partial y}-\beta^{\prime}x\frac{\partial{F_{1}}% }{\partial x}-\alpha\beta^{\prime}{F_{1}}=0,$

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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 with respect to and $\,\partial\NVar{x}$ : partial differential of
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

$x(1-x)\frac{{\partial}^{2}{F_{2}}}{{\partial x}^{2}}-xy\frac{\,{\partial}^{2}{% F_{2}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{% \partial{F_{2}}}{\partial x}-\beta y\frac{\partial{F_{2}}}{\partial y}-\alpha% \beta{F_{2}}=0,$

$y(1-y)\frac{{\partial}^{2}{F_{2}}}{{\partial y}^{2}}-xy\frac{\,{\partial}^{2}{% F_{2}}}{\,\partial x\,\partial y}+\left(\gamma^{\prime}-(\alpha+\beta^{\prime}% +1)y\right)\frac{\partial{F_{2}}}{\partial y}-\beta^{\prime}x\frac{\partial{F_% {2}}}{\partial x}-\alpha\beta^{\prime}{F_{2}}=0,$

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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 with respect to and $\,\partial\NVar{x}$ : partial differential of
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

$x(1-x)\frac{{\partial}^{2}{F_{3}}}{{\partial x}^{2}}+y\frac{\,{\partial}^{2}{F% _{3}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{% \partial{F_{3}}}{\partial x}-\alpha\beta{F_{3}}=0,$

$y(1-y)\frac{{\partial}^{2}{F_{3}}}{{\partial y}^{2}}+x\frac{\,{\partial}^{2}{F% _{3}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha^{\prime}+\beta^{\prime}+% 1)y\right)\frac{\partial{F_{3}}}{\partial y}-\alpha^{\prime}\beta^{\prime}{F_{% 3}}=0,$

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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 with respect to and $\,\partial\NVar{x}$ : partial differential of
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

$x(1-x)\frac{{\partial}^{2}{F_{4}}}{{\partial x}^{2}}-2xy\frac{\,{\partial}^{2}% {F_{4}}}{\,\partial x\,\partial y}-y^{2}\frac{{\partial}^{2}{F_{4}}}{{\partial y% }^{2}}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{\partial{F_{4}}}{\partial x}% -(\alpha+\beta+1)y\frac{\partial{F_{4}}}{\partial y}-\alpha\beta{F_{4}}=0,$

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

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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 with respect to and $\,\partial\NVar{x}$ : partial differential of
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 $G_{2}(\alpha,\alpha^{\prime};\beta,\beta^{\prime};x,y)=\sum_{m,n=0}^{\infty}% \frac{\Gamma\left(\alpha+m\right)\Gamma\left(\alpha^{\prime}+n\right)\Gamma% \left(\beta+n-m\right)\Gamma\left(\beta^{\prime}+m-n\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\alpha^{\prime}\right)\Gamma\left(\beta\right)\Gamma% \left(\beta^{\prime}\right)}\frac{x^{m}y^{n}}{m!n!},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function and : factorial (as in )
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 $G_{3}(\alpha,\alpha^{\prime};x,y)=\sum_{m,n=0}^{\infty}\frac{\Gamma\left(% \alpha+2n-m\right)\Gamma\left(\alpha^{\prime}+2m-n\right)}{\Gamma\left(\alpha% \right)\Gamma\left(\alpha^{\prime}\right)}\frac{x^{m}y^{n}}{m!n!},$ $|x|+|y|<\frac{1}{4}$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function and : factorial (as in )
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