DLMF:25.11.E45 (Q7719)

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DLMF:25.11.E45
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    Defining formula
    ζ ′ ⁡ ( - 2 , a ) - 1 12 ⁢ a + 1 9 ⁢ a 3 - ( 1 6 ⁢ a - 1 2 ⁢ a 2 + 1 3 ⁢ a 3 ) ⁢ ln ⁡ a ∼ ∑ k = 1 ∞ 2 ⁢ B 2 ⁢ k + 2 ( 2 ⁢ k + 2 ) ⁢ ( 2 ⁢ k + 1 ) ⁢ 2 ⁢ k ⁢ ( 2 ⁢ k - 1 ) ⁢ a - ( 2 ⁢ k - 1 ) . asymptotic-expansion diffop Hurwitz-zeta 1 2 𝑎 1 12 𝑎 1 9 superscript 𝑎 3 1 6 𝑎 1 2 superscript 𝑎 2 1 3 superscript 𝑎 3 𝑎 superscript subscript 𝑘 1 2 Bernoulli-number-B 2 𝑘 2 2 𝑘 2 2 𝑘 1 2 𝑘 2 𝑘 1 superscript 𝑎 2 𝑘 1 {\displaystyle{\displaystyle\zeta'\left(-2,a\right)-\frac{1}{12}a+\frac{1}{9}a% ^{3}-\left(\frac{1}{6}a-\frac{1}{2}a^{2}+\frac{1}{3}a^{3}\right)\ln a\sim\sum_% {k=1}^{\infty}\frac{2B_{2k+2}}{(2k+2)(2k+1)2k(2k-1)}a^{-(2k-1)}.}}
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    DLMF id
    DLMF:25.11.E45
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