DLMF:24.8.E8 (Q7495)

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DLMF:24.8.E8
No description defined

    Statements

    Defining formula
    B 2 ⁢ n 4 ⁢ n ⁢ ( α n - ( - β ) n ) = α n ⁢ ∑ k = 1 ∞ k 2 ⁢ n - 1 e 2 ⁢ α ⁢ k - 1 - ( - β ) n ⁢ ∑ k = 1 ∞ k 2 ⁢ n - 1 e 2 ⁢ β ⁢ k - 1 , Bernoulli-number-B 2 𝑛 4 𝑛 superscript 𝛼 𝑛 superscript 𝛽 𝑛 superscript 𝛼 𝑛 superscript subscript 𝑘 1 superscript 𝑘 2 𝑛 1 superscript 𝑒 2 𝛼 𝑘 1 superscript 𝛽 𝑛 superscript subscript 𝑘 1 superscript 𝑘 2 𝑛 1 superscript 𝑒 2 𝛽 𝑘 1 {\displaystyle{\displaystyle\frac{B_{2n}}{4n}\left(\alpha^{n}-(-\beta)^{n}% \right)=\alpha^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}-(-\beta)% ^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\beta k}-1},}}
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    DLMF id
    DLMF:24.8.E8
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    constraint
    n = 2 , 3 , … 𝑛 2 3 … {\displaystyle{\displaystyle n=2,3,\dots}}
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    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1abdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    Q12113
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C24.S1.XMD2.m1gdec
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    Q12110
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C24.S1.XMD5.m1gdec
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    Q12118
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C24.S8.XMD1.m1dec
    0 references
    Q12119
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C24.S8.XMD2.m1dec
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