§5.16 Sums

Notes:: See Jordan (1939, pp. 344–345).
Keywords:: polygamma functions
Permalink:: http://dlmf.nist.gov/5.16

5.16.1 $\sum _{{k=1}}^{\infty}(-1)^{k}{\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(k\right)=-\frac{\pi^{2}}{8},$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.16.E1
Encodings:: TeX, pMML, png

5.16.2 $\sum _{{k=1}}^{\infty}\frac{1}{k}{\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(k+1\right)=\mathop{\zeta\/}\nolimits\!\left(3\right)=-\frac{1}{2}{\mathop{\psi\/}\nolimits^{{\prime\prime}}}\!\left(1\right).$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.16.E2
Encodings:: TeX, pMML, png

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 §§15.2(i), 16.2.

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, pp. 90, 149).