DLMF:8.7.E2 (Q3226)

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DLMF:8.7.E2
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    Statements

    DLMF defining formula
    γ ⁡ ( a , x + y ) - γ ⁡ ( a , x ) = Γ ⁡ ( a , x ) - Γ ⁡ ( a , x + y ) = e - x ⁢ x a - 1 ⁢ ∑ n = 0 ∞ ( 1 - a ) n ( - x ) n ⁢ ( 1 - e - y ⁢ e n ⁢ ( y ) ) , incomplete-gamma 𝑎 𝑥 𝑦 incomplete-gamma 𝑎 𝑥 incomplete-Gamma 𝑎 𝑥 incomplete-Gamma 𝑎 𝑥 𝑦 superscript 𝑒 𝑥 superscript 𝑥 𝑎 1 superscript subscript 𝑛 0 Pochhammer 1 𝑎 𝑛 superscript 𝑥 𝑛 1 superscript 𝑒 𝑦 subscript 𝑒 𝑛 𝑦 {\displaystyle{\displaystyle\gamma\left(a,x+y\right)-\gamma\left(a,x\right)=% \Gamma\left(a,x\right)-\Gamma\left(a,x+y\right)=e^{-x}x^{a-1}\sum_{n=0}^{% \infty}\frac{{\left(1-a\right)_{n}}}{(-x)^{n}}(1-e^{-y}e_{n}(y)),}}
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    constraint
    | y | < | x | 𝑦 𝑥 {\displaystyle{\displaystyle|y|<|x|}}
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    Symbols used
    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
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    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    incomplete gamma function
    DLMF defining formula
    Γ ⁡ ( a , z ) incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E2.m2adec
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    incomplete gamma function
    DLMF defining formula
    γ ⁡ ( a , z ) incomplete-gamma 𝑎 𝑧 {\displaystyle{\displaystyle\gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E1.m2adec
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    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C8.S1.XMD1.m1dec
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    parameter
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1adec
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    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C8.S1.XMD6.m1dec
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    functions (locally)
    DLMF defining formula
    e n ⁢ ( z ) subscript 𝑒 𝑛 𝑧 {\displaystyle{\displaystyle e_{n}(z)}}
    xml-id
    C8.S4.XMD1.m1dec
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    Identifiers

     
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