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DLMF:16.16.E5 - MaRDI portal

DLMF:16.16.E5 (Q5968)

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DLMF:16.16.E5
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    Statements

    DLMF defining formula
    F 3 ⁡ ( α , γ - α ; β , γ - β ; γ ; x , y ) = ( 1 - y ) α + β - γ ⁢ F 1 2 ⁡ ( α , β γ ; x + y - x ⁢ y ) , Appell-F-3 𝛼 𝛾 𝛼 𝛽 𝛾 𝛽 𝛾 𝑥 𝑦 superscript 1 𝑦 𝛼 𝛽 𝛾 Gauss-hypergeometric-F-as-2F1 𝛼 𝛽 𝛾 𝑥 𝑦 𝑥 𝑦 {\displaystyle{\displaystyle{F_{3}}\left(\alpha,\gamma-\alpha;\beta,\gamma-% \beta;\gamma;x,y\right)=(1-y)^{\alpha+\beta-\gamma}{{}_{2}F_{1}}\left({\alpha,% \beta\atop\gamma};x+y-xy\right),}}
    0 references
    Symbols used
    third Appell function
    DLMF defining formula
    F 3 ⁡ ( α , α ′ ; β , β ′ ; γ ; x , y ) Appell-F-3 𝛼 superscript 𝛼 ′ 𝛽 superscript 𝛽 ′ 𝛾 𝑥 𝑦 {\displaystyle{\displaystyle{F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};% \NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)}}
    xml-id
    C16.S13.E3.m2aadec
    0 references
    = F ( a , b ; c ; z ) notation for Gauss’ hypergeometric function
    DLMF 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.m5abdec
    0 references

    Identifiers

     
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