A non-commutative binomial formula.
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.
scientific article; zbMATH DE number 1422967