How to add a non-integer number of terms, and how to produce unusual infinite summations
type: Item
instance of: scholarly article
title: How to add a non-integer number of terms, and how to produce unusual infinite summations
zbMATH Open document ID: 1064.40003
author: Dierk Schleicher
author: Markus Müller
published in: Journal of Computational and Applied Mathematics
publication date: 2005-04-18T00:00:00Z
review text: In the very interesting paper under review, the authors extend the definition of a sum \(\sum^x_{y=1} f(y)\) to the case when \(x\) is a real or complex number. The fractional sum for polynomials is defined by \(\sum^x_{k=1} p(k):= P(x)\) (where \(p:\mathbb{C}\to\mathbb{C}\) is the given polynomial), \(P\) denoting the unique polynomial that agrees with the classical value of the sum of every positive integer \(x\). For arbitrary \(a,b\in\mathbb{C}\), put \(\sum^b_{k=a} p(k)= P(b)- P(a-1)\). Let now \(U\subset\mathbb{C}\) and \(\sigma\in \mathbb{N}\cup\{-\infty\}\). A function \(f: U\to\mathbb{C}\) will be called right summable of degree \(\sigma\) if: (i) \(U+1\subset U\) (Minkowski summation); (ii) there exists a sequence of polynomials \((p_n)\) of fixed degree \(\sigma\) such that \(|f(n+x)- p_n(n+ x)|\to 0\) as \(n\to\infty\), for all \(x\) in every bounded subset of \(U\); (iii) for \(a,b+1\in U\), the limit \[ \lim_{n\to\infty} \Biggl(\sum^{n+b}_{k= n+a} p_n(k)\Biggr)+ \sum^n_{k=1} (f(k+a-1)- f(k+ b)) \] exists. (Here, unlike the authors, we preserved the same notation for fractional sums, which may be obviously distinguished from a usual sum.) Fractional products are defined by \[ \prod^b_{k=a} f(k)= \exp\Biggl(\sum^b_{k=a} \ln f(k)\Biggr). \] Left fractional sums or products may be defined in a simlar manner. By using some properties of fractional sums and products, the authors deduce in a unitary way certain unusual infinite sums, products, and limits. They state that, while this method has been mentioned already by Euler and Ramanujan, no systematic attempts to work it out were known up to now.
reviewed by: József Sándor
Mathematics Subject Classification ID: 40A25
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 33C99
zbMATH DE Number: 2157679
zbMATH Keywords: summation of series
zbMATH Keywords: interpolation
zbMATH Keywords: fractional sums
MaRDI profile type: Publication
full work available at URL: j.cam.2004.08.009
OpenAlex ID: W2078397539
cites work: Q4222737
cites work: Q3785143
cites work: On some strange summation formulas
DOI: 10.1016/J.CAM.2004.08.009
description: scientific article; zbMATH DE number 2157679