DLMF:16.16.E7 (Q5273)

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DLMF:16.16.E7
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    Statements

    Defining formula
    F 4 ⁡ ( α , β ; γ , γ ′ ; x ⁢ ( 1 - y ) , y ⁢ ( 1 - x ) ) = ∑ k = 0 ∞ ( α ) k ⁢ ( β ) k ⁢ ( α + β - γ - γ ′ + 1 ) k ( γ ) k ⁢ ( γ ′ ) k ⁢ k ! ⁢ x k ⁢ y k ⁢ F 1 2 ⁡ ( α + k , β + k γ + k ; x ) ⁢ F 1 2 ⁡ ( α + k , β + k γ ′ + k ; y ) ; Appell-F-4 𝛼 𝛽 𝛾 superscript 𝛾 ′ 𝑥 1 𝑦 𝑦 1 𝑥 superscript subscript 𝑘 0 Pochhammer 𝛼 𝑘 Pochhammer 𝛽 𝑘 Pochhammer 𝛼 𝛽 𝛾 superscript 𝛾 ′ 1 𝑘 Pochhammer 𝛾 𝑘 Pochhammer superscript 𝛾 ′ 𝑘 𝑘 superscript 𝑥 𝑘 superscript 𝑦 𝑘 Gauss-hypergeometric-F-as-2F1 𝛼 𝑘 𝛽 𝑘 𝛾 𝑘 𝑥 Gauss-hypergeometric-F-as-2F1 𝛼 𝑘 𝛽 𝑘 superscript 𝛾 ′ 𝑘 𝑦 {\displaystyle{\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x% (1-y),y(1-x)\right)=\sum_{k=0}^{\infty}\frac{{\left(\alpha\right)_{k}}{\left(% \beta\right)_{k}}{\left(\alpha+\beta-\gamma-\gamma^{\prime}+1\right)_{k}}}{{% \left(\gamma\right)_{k}}{\left(\gamma^{\prime}\right)_{k}}k!}x^{k}y^{k}{{}_{2}% F_{1}}\left({\alpha+k,\beta+k\atop\gamma+k};x\right){{}_{2}F_{1}}\left({\alpha% +k,\beta+k\atop\gamma^{\prime}+k};y\right);}}
    0 references
    DLMF id
    DLMF:16.16.E7
    0 references
    Symbols used
    fourth Appell function
    Defining formula
    F 4 ⁡ ( α , β ; γ , γ ′ ⁢ missing ⁢ missing ; missing , missing ) Appell-F-4 𝛼 𝛽 𝛾 superscript 𝛾 ′ missing missing {\displaystyle{\displaystyle{F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{% \gamma},\NVar{\gamma^{\prime}}{missing}{missing};missing,missing\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C16.S13.E4.m2aadec
    0 references
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5addec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
     
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