DLMF:13.7.E11 (Q4423)

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DLMF:13.7.E11
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    Statements

    Defining formula
    R n ⁢ ( a , b , z ) = ( - 1 ) n ⁢ 2 ⁢ π ⁢ z a - b Γ ⁡ ( a ) ⁢ Γ ⁡ ( a - b + 1 ) ⁢ ( ∑ s = 0 m - 1 ( 1 - a ) s ⁢ ( b - a ) s s ! ⁢ ( - z ) - s ⁢ G n + 2 ⁢ a - b - s ⁢ ( z ) + ( 1 - a ) m ⁢ ( b - a ) m ⁢ R m , n ⁢ ( a , b , z ) ) , subscript 𝑅 𝑛 𝑎 𝑏 𝑧 superscript 1 𝑛 2 𝜋 superscript 𝑧 𝑎 𝑏 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑏 1 superscript subscript 𝑠 0 𝑚 1 Pochhammer 1 𝑎 𝑠 Pochhammer 𝑏 𝑎 𝑠 𝑠 superscript 𝑧 𝑠 subscript 𝐺 𝑛 2 𝑎 𝑏 𝑠 𝑧 Pochhammer 1 𝑎 𝑚 Pochhammer 𝑏 𝑎 𝑚 subscript 𝑅 𝑚 𝑛 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle R_{n}(a,b,z)=\frac{(-1)^{n}2\pi z^{a-b}}{\Gamma% \left(a\right)\Gamma\left(a-b+1\right)}\left(\sum_{s=0}^{m-1}\frac{{\left(1-a% \right)_{s}}{\left(b-a\right)_{s}}}{s!}(-z)^{-s}G_{n+2a-b-s}(z)+{\left(1-a% \right)_{m}}{\left(b-a\right)_{m}}R_{m,n}(a,b,z)\right),}}
    0 references
    DLMF id
    DLMF:13.7.E11
    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.m2acdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5afdec
    0 references
     
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