DLMF:15.2.E1 (Q4977)

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DLMF:15.2.E1
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    Statements

    Defining formula
    F ⁡ ( a , b ; c ; z ) = ∑ s = 0 ∞ ( a ) s ⁢ ( b ) s ( c ) s ⁢ s ! ⁢ z s = 1 + a ⁢ b c ⁢ z + a ⁢ ( a + 1 ) ⁢ b ⁢ ( b + 1 ) c ⁢ ( c + 1 ) ⁢ 2 ! ⁢ z 2 + ⋯ = Γ ⁡ ( c ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ ∑ s = 0 ∞ Γ ⁡ ( a + s ) ⁢ Γ ⁡ ( b + s ) Γ ⁡ ( c + s ) ⁢ s ! ⁢ z s , Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑠 Pochhammer 𝑏 𝑠 Pochhammer 𝑐 𝑠 𝑠 superscript 𝑧 𝑠 1 𝑎 𝑏 𝑐 𝑧 𝑎 𝑎 1 𝑏 𝑏 1 𝑐 𝑐 1 2 superscript 𝑧 2 ⋯ Euler-Gamma 𝑐 Euler-Gamma 𝑎 Euler-Gamma 𝑏 superscript subscript 𝑠 0 Euler-Gamma 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 Euler-Gamma 𝑐 𝑠 𝑠 superscript 𝑧 𝑠 {\displaystyle{\displaystyle F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{% \left(a\right)_{s}}{\left(b\right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}=1+\frac{% ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)2!}z^{2}+\cdots=\frac{\Gamma\left(c\right)}{% \Gamma\left(a\right)\Gamma\left(b\right)}\sum_{s=0}^{\infty}\frac{\Gamma\left(% a+s\right)\Gamma\left(b+s\right)}{\Gamma\left(c+s\right)s!}z^{s},}}
    0 references
    DLMF id
    DLMF:15.2.E1
    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
    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.m5adec
    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
    Q11648
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C15.S1.XMD11.m1dec
    0 references
     
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