DLMF: GA.16 Sums

§GA.16. Sums

Sources:

Jordan(1939)(pp. 344–345).

Notes:

Jordan(1939)(pp. 344–345)

GA.16.1 $\sum _{{k=1}}^{\infty}(-1)^{k}{{\psi}^{{\prime}}}\!\left(k\right)=-\frac{\pi^{2}}{8},$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function and $k$ : nonnegative integer
Encodings:: pMathML, png, TeX

GA.16.2 $\sum _{{k=1}}^{\infty}\frac{1}{k}{{\psi}^{{\prime}}}\!\left(k+1\right)=\zeta\!\left(3\right)=-\frac{1}{2}{{\psi}^{{\prime\prime}}}\!\left(1\right).$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function and $k$ : nonnegative integer
Encodings:: pMathML, png, TeX

For further sums involving the psi function see Hansen(1975)(pp. 360–367). For sums of gamma functions see Andrews et.al.(1999)(Chapters 2 and 3) and §§, .

For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et.al.(1999)(Chapter 1) and Terras(1999).