16 Generalized Hypergeometric Functions & Meijer G-Function Two-Variable Hypergeometric Functions 16.12 Products 16.14 Partial Differential Equations

§16.13 Appell Functions

ⓘ

Keywords:: Appell functions, definition, hypergeometric function, notation, with two variables
Notes:: See Appell and Kampé de Fériet (1926) and Erdélyi et al. (1953a, §5.7).
Referenced by:: §19.25(vii), §19.5, Erratum (V1.0.10) for References, Erratum (V1.0.11) for References, Erratum (V1.0.9) for References
Permalink:: http://dlmf.nist.gov/16.13
Addition (effective with 1.0.11):: The reference Appell and Kampé de Fériet (1926) was added to the notes.
Addition (effective with 1.0.10):: References to López et al. (2013a) and Ferreira et al. (2013b, a) have been added at the end of this section.
Addition (effective with 1.0.9):: The reference to López et al. (2013b) has been added at the end of this section.
See also:: Annotations for Ch.16

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

16.13.1	$\displaystyle{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$	$\displaystyle=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(% \beta\right)_{m}}{\left(\beta^{\prime}\right)_{n}}}{{\left(\gamma\right)_{m+n}% }m!n!}x^{m}y^{n},$
$\max\left(\|x\|,\|y\|\right)<1$ ,
ⓘ Defines: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ ) Permalink: http://dlmf.nist.gov/16.13.E1 Encodings: TeX, pMML, png See also: Annotations for §16.13 and Ch.16
16.13.2	$\displaystyle{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x% ,y\right)$	$\displaystyle=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(% \beta\right)_{m}}{\left(\beta^{\prime}\right)_{n}}}{{\left(\gamma\right)_{m}}{% \left(\gamma^{\prime}\right)_{n}}m!n!}x^{m}y^{n},$
$\|x\|+\|y\|<1$ ,
ⓘ Defines: ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ ) Permalink: http://dlmf.nist.gov/16.13.E2 Encodings: TeX, pMML, png See also: Annotations for §16.13 and Ch.16
16.13.3	$\displaystyle{F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x% ,y\right)$	$\displaystyle=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m}}{\left(% \alpha^{\prime}\right)_{n}}{\left(\beta\right)_{m}}{\left(\beta^{\prime}\right% )_{n}}}{{\left(\gamma\right)_{m+n}}m!n!}x^{m}y^{n},$
$\max\left(\|x\|,\|y\|\right)<1$ ,
ⓘ Defines: ${F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$ : third Appell function Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ ) Permalink: http://dlmf.nist.gov/16.13.E3 Encodings: TeX, pMML, png See also: Annotations for §16.13 and Ch.16
16.13.4	$\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)$	$\displaystyle=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(% \beta\right)_{m+n}}}{{\left(\gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n% }}m!n!}x^{m}y^{n},$
$\sqrt{\|x\|}+\sqrt{\|y\|}<1$ .
ⓘ Defines: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ ) Permalink: http://dlmf.nist.gov/16.13.E4 Encodings: TeX, pMML, png See also: Annotations for §16.13 and Ch.16

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

For large parameter asymptotics see López et al. (2013a, b), and Ferreira et al. (2013a, b).

16.12 Products 16.14 Partial Differential Equations

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