On a queer binomial sum
type: Item
instance of: scholarly article
title: On a queer binomial sum
zbMATH Open document ID: 0913.11010
author: Richard Warlimont
author: R. Denk
published in: Archiv der Mathematik
publication date: 1999-04-11T00:00:00Z
full work available at URL: urn:nbn:de:bsz:352-opus-50740
review text: The authors consider the binomial sum \[ S_n(r)=\sum_{m=0}^n{n\choose m}\frac{(-1)^m}{m^r+1} \] with \(r>0\), \(n\in{\mathbb N}\). It turns out that the behaviour of this sum for \(n\to\infty\) depends on the parameter \(r\) and changes dramatically at the values \(r=1\) and \(r=2\). In particular, for \(r\geq 2\) an oscillatory behaviour occurs while for \(r<2\) finally the sequence \(S_n(r)\) becomes monotonous. To be more precise (see Theorem 2), \(S_n(r)\sim\Gamma(r)(\log n)^{-r}\) for \(0<r<1\), \(S_n(r)\sim-\Gamma(r)(\log n)^{-r}\) for \(1<r<2\), and for \(r\geq 2\) it holds that \[ S_n(r)=n^{\cos(\pi/r)}c(r)\cos[a(r)\log n+b(r)]+O(n^{-\delta(r)}) \] with explicitly determined terms \(a(r)\), \(b(r)\), \(c(r)\) and \(\delta(r)\), which are too complicated to be given here. For \(r=1\) the surprisingly simple equation \(S_n(1)=\frac 1{n+1}\) holds. The proofs were established by using Fourier integrals including complex integration depending on the analytical behavior of \(\frac 1{z^r+1}\) for \(z\in{\mathbb C}\).
reviewed by: Helmut Müller
Mathematics Subject Classification ID: 11B65
zbMATH DE Number: 1203766
zbMATH Keywords: binomial sum
zbMATH Keywords: Fourier integrals
MaRDI profile type: Publication
OpenAlex ID: W2154642514
DOI: 10.1007/S000130050230
Recommended article: Q5753955
Recommended article: ON SUMS OF BINOMIAL COEFFICIENTS
Recommended article: Congruences on sums of \(q\)-binomial coefficients
Recommended article: Q3428608
Recommended article: Q5033182
Recommended article: Q5464267
Recommended article: Q3628987
Recommended article: On some congruences of certain binomial sums
Recommended article: Q3791457
Recommended article: Q5330705
description: scientific article; zbMATH DE number 1203766