5 Gamma Function Properties 5.15 Polygamma Functions 5.17 Barnes’

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§5.16 Sums

ⓘ

Keywords:: polygamma functions, sums
Notes:: See Jordan (1939, pp. 344–345).
Permalink:: http://dlmf.nist.gov/5.16
See also:: Annotations for Ch.5

5.16.1	$\displaystyle\sum_{k=1}^{\infty}(-1)^{k}\psi'\left(k\right)$	$\displaystyle=-\frac{\pi^{2}}{8},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/5.16.E1 Encodings: TeX, pMML, png See also: Annotations for §5.16 and Ch.5
5.16.2	$\displaystyle\sum_{k=1}^{\infty}\frac{1}{k}\psi'\left(k+1\right)$	$\displaystyle=\zeta\left(3\right)=-\frac{1}{2}\psi''\left(1\right).$
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/5.16.E2 Encodings: TeX, pMML, png See also: Annotations for §5.16 and Ch.5

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).

5.15 Polygamma Functions 5.17 Barnes’

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-Function (Double Gamma Function)

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