Digital Library of Mathematical Functions
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16 Generalized Hypergeometric Functions and Meijer G-FunctionTwo-Variable Hypergeometric Functions

§16.13 Appell Functions

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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,
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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,
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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,
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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.
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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.

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