Summation of series and Gaussian quadratures. II
type: Item
instance of: scholarly article
title: Summation of series and Gaussian quadratures. II
zbMATH Open document ID: 0827.40001
DOI: 10.1007/BF02198299
published in: Numerical Algorithms
publication date: 1995-11-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 40A25
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 784684
zbMATH Keywords: summation of series
zbMATH Keywords: Gaussian quadratures
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: convergent series
zbMATH Keywords: hypergeometric functions
Wikidata QID: Q115058808
author: Gradimir V. Milovanović
reviewed by: Jürgen Müller
MaRDI profile type: Publication
cites work: Q5558293
cites work: A Class of Slowly Convergent Series and their Summation by Gaussian Quadrature
cites work: Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series
cites work: Q4079940
cites work: Q4749793
full work available at URL: https://doi.org/10.1007/bf02198299
OpenAlex ID: W2143571792
description: scientific article; zbMATH DE number 784684