The paper continues previous work of the author [Summation of series and Gaussian quadratures. In: Zahar, R. V. M. (ed.): Approximation and computation: a Festschrift in honor of Walter Gautschi, Boston, US: Birkhäuser, ISNM 119, 459-475 (1994; Zbl 0816.41027)] concerning the summation of some classes of slowly convergent series. The series to be considered here are of the form \(\sum_{k=m}^\infty (\pm 1)^k k^{\nu-1}/ (k+ a)^p\), where \(m\in \mathbb{Z}\), \(0< \nu\leq 1\) and \(a\) and \(p\) are such that convergence takes place. For \(m= \nu= 1\) and \(a=0\) we recover the Riemann \(\zeta\)-function.
The idea is to express the series as a finite sum plus a weighted integral over \((0, \infty)\) and then to apply Gaussian quadrature to be integral. For the above series, the integrands may be expressed in terms of hypergeometric functions. Several numerical examples illustrate the effectivity of the method.