DLMF:24.4.E11 (Q7424)

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DLMF:24.4.E11
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    Statements

    Defining formula
    ∑ k = 1 ( k , m ) = 1 m k n = 1 n + 1 ⁢ ∑ j = 1 n + 1 ( n + 1 j ) ⁢ ( ∏ p | m ( 1 - p n - j ) ⁢ B n + 1 - j ) ⁢ m j . superscript subscript 𝑘 1 𝑘 𝑚 1 𝑚 superscript 𝑘 𝑛 1 𝑛 1 superscript subscript 𝑗 1 𝑛 1 binomial 𝑛 1 𝑗 subscript product divides 𝑝 𝑚 1 superscript 𝑝 𝑛 𝑗 Bernoulli-number-B 𝑛 1 𝑗 superscript 𝑚 𝑗 {\displaystyle{\displaystyle\sum_{\begin{subarray}{c}k=1\\ \left(k,m\right)=1\end{subarray}}^{m}k^{n}=\frac{1}{n+1}\sum_{j=1}^{n+1}{n+1% \choose j}\*\left(\prod_{p\mathbin{|}m}(1-p^{n-j})B_{n+1-j}\right)m^{j}.}}
    0 references
    DLMF id
    DLMF:24.4.E11
    0 references
    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1aadec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1adec
    0 references
    Q12093
    Defining formula
    ( m , n ) 𝑚 𝑛 {\displaystyle{\displaystyle\left(\NVar{m},\NVar{n}\right)}}
    xml-id
    C27.S1.p2.t1.r3.m6adec
    0 references
    Q12116
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C24.S1.XMD1.m1dec
    0 references
    Q12113
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C24.S1.XMD2.m1ddec
    0 references
     
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