§16.13 Appell Functions

Notes:: See Erdélyi et al. (1953a, §5.7).
Keywords:: Appell functions, hypergeometric function
Referenced by:: §19.25(vii), §19.5
Permalink:: http://dlmf.nist.gov/16.13

The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1):

16.13.1 $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)=\sum _{{m,n=0}}^{\infty}\frac{\left(\alpha\right)_{{m+n}}\left(\beta\right)_{{m}}\left(\beta^{{\prime}}\right)_{{n}}}{\left(\gamma\right)_{{m+n}}m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$ ,

Defines:: $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$ : Appell function
Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial and $(a,b)$ : open interval
Permalink:: http://dlmf.nist.gov/16.13.E1
Encodings:: TeX, pMML, png

16.13.2 $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,\gamma^{{\prime}};x,y\right)=\sum _{{m,n=0}}^{\infty}\frac{\left(\alpha\right)_{{m+n}}\left(\beta\right)_{{m}}\left(\beta^{{\prime}}\right)_{{n}}}{\left(\gamma\right)_{{m}}\left(\gamma^{{\prime}}\right)_{{n}}m!n!}x^{m}y^{n},$ $|x|+|y|<1$ ,

Defines:: $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,\gamma^{{\prime}};x,y\right)$ : Appell function
Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol and $!$ : $n!$ : factorial
Permalink:: http://dlmf.nist.gov/16.13.E2
Encodings:: TeX, pMML, png

16.13.3 $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\alpha^{{\prime}};\beta,\beta^{{\prime}};\gamma;x,y\right)=\sum _{{m,n=0}}^{\infty}\frac{\left(\alpha\right)_{{m}}\left(\alpha^{{\prime}}\right)_{{n}}\left(\beta\right)_{{m}}\left(\beta^{{\prime}}\right)_{{n}}}{\left(\gamma\right)_{{m+n}}m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$ ,

Defines:: $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\alpha^{{\prime}};\beta,\beta^{{\prime}};\gamma;x,y\right)$ : Appell function
Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial and $(a,b)$ : open interval
Permalink:: http://dlmf.nist.gov/16.13.E3
Encodings:: TeX, pMML, png

16.13.4 $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y\right)=\sum _{{m,n=0}}^{\infty}\frac{\left(\alpha\right)_{{m+n}}\left(\beta\right)_{{m+n}}}{\left(\gamma\right)_{{m}}\left(\gamma^{{\prime}}\right)_{{n}}m!n!}x^{m}y^{n},$ $\sqrt{|x|}+\sqrt{|y|}<1$ .

Defines:: $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y\right)$ : Appell function
Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol and $!$ : $n!$ : factorial
Permalink:: http://dlmf.nist.gov/16.13.E4
Encodings:: TeX, pMML, png

Here and elsewhere it is assumed that neither of the bottom parameters $\gamma$ and $\gamma^{{\prime}}$ is a nonpositive integer.