A binomial sum related to Wolstenholme's theorem
type: Item
instance of: scholarly article
title: A binomial sum related to Wolstenholme's theorem
zbMATH Open document ID: 1241.11016
author: Marc Chamberland
author: Karl Dilcher
published in: Journal of Number Theory
publication date: 2009-09-15T00:00:00Z
review text: \textit{M. Chamberland} and \textit{K. Dilcher} [J. Number Theory 120, No. 2, 349--371 (2006; Zbl 1139.11013)] had already found that a certain alternating sum \(u(n)\) of \(n+1\) products of two binomial coefficients has a property similar to Wolstenholme's theorem, namely \(u(p)\equiv -\pmod {p^3}\) for all primes \(p\geq 5\). In order to partly explain why such congruence also holds for certain composite integers \(p\) which appear to always have exactly two prime divisors, one of which is always 2 or 5, in the present paper the authors study the sums \[ u(n):=\sum_{k=0}^n (-1)^{k} \binom{n}{k} \binom{2n}{k} \] in greater details than they previously did. As a consequence of their accurate research, the composites in question are characterized and some new open problems are raised as well. Beyond basic tools like Leibniz's rule for higher derivatives of a product, Fermat's Little Theorem, Kummer's Theorem, Vandermonde's Identity and the Chinese Remainder Theorem, in the proof the authors employ some remarkable results for binomial coefficients from \textit{A. Granville} [Borwein, J. (ed.) et al., Organic mathematics. CMS Conf. Proc. 20, 253--276 (1997; Zbl 0903.11005)], the famous congruence discovered by \textit{H. Anton} [Arch. Math. Phys. 49, 241--308 (1868; JFM 01.0047.04)], an important congruence obtained by \textit{V. Brun} et al. [11. Skand. Mat.-Kongr., Trondheim 1949, 42--54 (1952; Zbl 0048.27204)] and generalized by \textit{K. Davis} and \textit{W. Webb} [J. Number Theory 43, No. 1, 20--23 (1993; Zbl 0769.11008)], a well-known theorem by \textit{E. Lucas} [Bull. Soc. Math. Fr. 6, 49--54 (1878; JFM 10.0139.04)], a convergence property of linear recurrence sequences found by \textit{R. J. Kooman} and \textit{R. Tijdeman} [Nieuw Arch. Wiskd., IV. Ser. 8, No. 1, 13--25 (1990; Zbl 0713.11010)], an explicit formula for the Jacobi polynomials from [\textit{H. W. Gould}, Combinatorial identities. Morgantown, W. Va.: Henry E. Gould (1972; Zbl 0241.05011)] and a…
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 5603992
zbMATH Keywords: binomial coefficients
zbMATH Keywords: binomial sums
zbMATH Keywords: Wolstenholme's theorem
zbMATH Keywords: congruences
zbMATH Keywords: Wilf-Zeilberger method
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.05.010
OpenAlex ID: W2015497605
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DOI: 10.1016/J.JNT.2009.05.010
description: scientific article; zbMATH DE number 5603992