§16.14 Partial Differential Equations

Permalink:: http://dlmf.nist.gov/16.14

§16.14(i) Appell Functions
§16.14(ii) Other Functions

§16.14(i) Appell Functions

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

16.14.1

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

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

Symbols:: $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$ : 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

16.14.2

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

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

Symbols:: $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,\gamma^{{\prime}};x,y\right)$ : 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

16.14.3

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

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

Symbols:: $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\alpha^{{\prime}};\beta,\beta^{{\prime}};\gamma;x,y\right)$ : 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

16.14.4

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

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

Symbols:: $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y\right)$ : 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

§16.14(ii) Other Functions

Permalink:: http://dlmf.nist.gov/16.14.ii

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 $G_{2}(\alpha,\alpha^{{\prime}};\beta,\beta^{{\prime}};x,y)=\sum _{{m,n=0}}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(\alpha+m\right)\mathop{\Gamma\/}\nolimits\!\left(\alpha^{{\prime}}+n\right)\mathop{\Gamma\/}\nolimits\!\left(\beta+n-m\right)\mathop{\Gamma\/}\nolimits\!\left(\beta^{{\prime}}+m-n\right)}{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(\alpha^{{\prime}}\right)\mathop{\Gamma\/}\nolimits\!\left(\beta\right)\mathop{\Gamma\/}\nolimits\!\left(\beta^{{\prime}}\right)}\frac{x^{m}y^{n}}{m!n!},$ $|x|<1$ , $|y|<1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $!$ : $n!$ : factorial
Permalink:: http://dlmf.nist.gov/16.14.E5
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $!$ : $n!$ : factorial
Permalink:: http://dlmf.nist.gov/16.14.E6
Encodings:: TeX, pMML, png

(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.

§16.14 Partial Differential Equations

Contents

§16.14(i) Appell Functions

§16.14(ii) Other Functions