DLMF:15.1.E2 (Q4976)

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DLMF:15.1.E2
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    Statements

    Defining formula
    F ⁡ ( a , b ; c ; z ) Γ ⁡ ( c ) = 𝐅 ⁡ ( a , b ; c ; z ) = 𝐅 ⁡ ( a , b c ; z ) = 𝐅 1 2 ⁡ ( a , b ; c ; z ) , Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Euler-Gamma 𝑐 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 hypergeometric-bold-pFq 2 1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}% =\mathbf{F}\left(a,b;c;z\right)=\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{% \mathbf{F}}_{1}}\left(a,b;c;z\right),}}
    0 references
    DLMF id
    DLMF:15.1.E2
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2aadec
    0 references
    scaled (or Olver’s) generalized hypergeometric function
    Defining formula
    𝐅 q p ⁡ ( 𝐚 ; 𝐛 ; ) hypergeometric-bold-pFq 𝑝 𝑞 𝐚 𝐛 {\displaystyle{\displaystyle{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{% \mathbf{a}};\NVar{\mathbf{b}};\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C16.S2.E5.m2adec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2adec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1adec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1adec
    0 references
     
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