DLMF:15.4.E22 (Q5006)

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DLMF:15.4.E22
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    Statements

    Defining formula
    lim z → 1 - ⁡ ( 1 - z ) a + b - c ⁢ ( F ⁡ ( a , b ; c ; z ) - Γ ⁡ ( c ) ⁢ Γ ⁡ ( c - a - b ) Γ ⁡ ( c - a ) ⁢ Γ ⁡ ( c - b ) ) = Γ ⁡ ( c ) ⁢ Γ ⁡ ( a + b - c ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) . subscript → 𝑧 limit-from 1 superscript 1 𝑧 𝑎 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Euler-Gamma 𝑐 Euler-Gamma 𝑐 𝑎 𝑏 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑐 𝑏 Euler-Gamma 𝑐 Euler-Gamma 𝑎 𝑏 𝑐 Euler-Gamma 𝑎 Euler-Gamma 𝑏 {\displaystyle{\displaystyle\lim_{z\to 1-}(1-z)^{a+b-c}\left(F\left(a,b;c;z% \right)-\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a% \right)\Gamma\left(c-b\right)}\right)=\frac{\Gamma\left(c\right)\Gamma\left(a+% b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}.}}
    0 references
    DLMF id
    DLMF:15.4.E22
    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.m2abdec
    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.m2audec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1tdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1pdec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1ddec
    0 references
     
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