On the {\(q\)}-analogue of the sum of cubes
type: Item
instance of: scholarly article
title: On the {\(q\)}-analogue of the sum of cubes
zbMATH Open document ID: 1071.05010
author: S. Ole Warnaar
published in: The Electronic Journal of Combinatorics
publication date: 2005-04-25T00:00:00Z
full work available at URL: https://eudml.org/doc/125074
full work available at URL: v11i1n13.html
review text: The sum, \(\sum_{k=1}^{n} k^3\), of the first \(n\) consecutive cubes is equal to \({{n+1}\choose{2}}^2\). The author gives a \(q\)-analogue of this result, \[ \sum_{k=1}^{n}q^{2n-2k} \frac{(1-q^k)^2(1-q^{2k})}{(1-q)^2(1-q^2)}=\left[ \begin{matrix} n+1\\ 2 \end{matrix} \right]^2 \] where \(\left[ \begin{matrix} n\\ k \end{matrix} \right] = \frac{(1-q^{n-k+1})(1-q^{n-k+2})\cdots(1-q^n)}{(1-q)(1-q^2)\cdots (1-q^k)}\), which is a simpler \(q\)-analogue than that discovered by [\textit{K. C. Garrett} and \textit{K. Hummel} [A combinatorial proof of the sum of \(q\)-cubes, Electron. J. Comb. 11, Research paper R9 (2004; Zbl 1050.05012)].
reviewed by: Angèle. M. Hamel
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05A30
zbMATH DE Number: 2161274
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2161274