DLMF:25.11.E21 (Q7695)

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DLMF:25.11.E21
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    test
    ζ ′ ⁡ ( 1 - 2 ⁢ n , h k ) = ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ⁢ k ) ) ⁢ B 2 ⁢ n ⁡ ( h / k ) 2 ⁢ n - ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ) ) ⁢ B 2 ⁢ n 2 ⁢ n ⁢ k 2 ⁢ n + ( - 1 ) n + 1 ⁢ π ( 2 ⁢ π ⁢ k ) 2 ⁢ n ⁢ ∑ r = 1 k - 1 sin ⁡ ( 2 ⁢ π ⁢ r ⁢ h k ) ⁢ ψ ( 2 ⁢ n - 1 ) ⁡ ( r k ) + ( - 1 ) n + 1 ⁢ 2 ⋅ ( 2 ⁢ n - 1 ) ! ( 2 ⁢ π ⁢ k ) 2 ⁢ n ⁢ ∑ r = 1 k - 1 cos ⁡ ( 2 ⁢ π ⁢ r ⁢ h k ) ⁢ ζ ′ ⁡ ( 2 ⁢ n , r k ) + ζ ′ ⁡ ( 1 - 2 ⁢ n ) k 2 ⁢ n , diffop Hurwitz-zeta 1 1 2 𝑛 ℎ 𝑘 digamma 2 𝑛 2 𝜋 𝑘 Bernoulli-polynomial-B 2 𝑛 ℎ 𝑘 2 𝑛 digamma 2 𝑛 2 𝜋 Bernoulli-number-B 2 𝑛 2 𝑛 superscript 𝑘 2 𝑛 superscript 1 𝑛 1 𝜋 superscript 2 𝜋 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝜋 𝑟 ℎ 𝑘 digamma 2 𝑛 1 𝑟 𝑘 ⋅ superscript 1 𝑛 1 2 2 𝑛 1 superscript 2 𝜋 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝜋 𝑟 ℎ 𝑘 diffop Hurwitz-zeta 1 2 𝑛 𝑟 𝑘 diffop Riemann-zeta 1 1 2 𝑛 superscript 𝑘 2 𝑛 {\displaystyle{\displaystyle\zeta'\left(1-2n,\frac{h}{k}\right)=\frac{(\psi% \left(2n\right)-\ln\left(2\pi k\right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi% \left(2n\right)-\ln\left(2\pi\right))B_{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2% \pi k)^{2n}}\sum_{r=1}^{k-1}\sin\left(\frac{2\pi rh}{k}\right){\psi^{(2n-1)}}% \left(\frac{r}{k}\right)+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=% 1}^{k-1}\cos\left(\frac{2\pi rh}{k}\right)\zeta'\left(2n,\frac{r}{k}\right)+% \frac{\zeta'\left(1-2n\right)}{k^{2n}},}}
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    DLMF id
    DLMF:25.11.E21
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    Symbols used
    Q11109
    test
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1addec
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