A non-commutative binomial formula.
type: Item
instance of: scholarly article
title: A non-commutative binomial formula.
zbMATH Open document ID: 1043.33011
DOI: 10.1016/S0393-0440(99)00032-7
author: Hjalmar Rosengren
published in: Journal of Geometry and Physics
publication date: 2000-05-29T00:00:00Z
review text: The classical binomial formula \[ (x+y)^n=\sum_{k=0}^n\binom{n}{k}y^kx^{n-k},\quad n=0,1,2,\ldots \] holds in an arbitrary ring with unity provided that \(xy=yx\). In [\textit{M. P. Schützenberger}, C.R. Acad. Sci., Paris 236, 352-353 (1953; Zbl 0051.09401)] it is shown that \[ (x+y)^n=\sum_{k=0}^n\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}y^kx^{n-k}, \quad n=0,1,2,\ldots, \] where \[ (a;q)_0=1\quad{\text and}\quad(a;q)_k:=\prod_{i=0}^{k-1}(1-aq^i), \quad k=1,2,3,\ldots \] provided that \(xy=qyx\). This noncommutative version of the binomial formula has been generalized by \textit{H.B. Benaoum} [J. Phys. A, Math. Gen. 32 , 2037--2040 (1999; Zbl 1042.33501)] for the case that \(xy=qyx+hy^2\) where \(h\) denotes another scalar. In the paper under review the author derives and proves a binomial formula for variables \(x\) and \(y\) satisfying the more general quadratic relation \(xy=ax^2+qyx+by^2\). Here \(x\) and \(y\) are elements of a ring with unity and \(a\), \(b\) and \(q\) are central elements.
Mathematics Subject Classification ID: 33D80
Mathematics Subject Classification ID: 58B32
zbMATH DE Number: 1422967
zbMATH Keywords: binomial formula
zbMATH Keywords: quadratic algebras
reviewed by: Roelof Koekoek
MaRDI profile type: Publication
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full work available at URL: s0393-0440(99)00032-7
OpenAlex ID: W2087983766
description: scientific article; zbMATH DE number 1422967