DLMF:8.8.E7 (Q3237)

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DLMF:8.8.E7
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    Statements

    DLMF defining formula
    γ ⁡ ( a + n , z ) = ( a ) n ⁢ γ ⁡ ( a , z ) - z a ⁢ e - z ⁢ ∑ k = 0 n - 1 Γ ⁡ ( a + n ) Γ ⁡ ( a + k + 1 ) ⁢ z k , incomplete-gamma 𝑎 𝑛 𝑧 Pochhammer 𝑎 𝑛 incomplete-gamma 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 superscript subscript 𝑘 0 𝑛 1 Euler-Gamma 𝑎 𝑛 Euler-Gamma 𝑎 𝑘 1 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\gamma\left(a+n,z\right)={\left(a\right)_{n}}% \gamma\left(a,z\right)-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a+n\right)% }{\Gamma\left(a+k+1\right)}z^{k},}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
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    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    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.m2aadec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1fdec
    0 references
    parameter
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1fdec
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    nonnegative integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1dec
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    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C8.S1.XMD6.m1dec
    0 references

    Identifiers

     
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