AI 中文总结
研究\((p,q)\) - 变形广义外尔代数中由变量\(X\)、\(Y\)和\(Z_p\)生成的非交换二项式公式\((X + Y)^n\)及相关恒等式，核心方法是用\((p,q)\) - 变形的\(s\) - 车数表示正规序系数，处理特殊情况得出新结果。
AI 中文摘要
我们研究由变量\(X\)、\(Y\)和\(Z_p\)生成的\((p,q)\) - 变形广义外尔代数，它们满足\((p,q)\) - 交换关系\(XY - qYX = hY^sZ_p\)，\(XZ_p = pZ_pX\)以及\(Z_pY = pYZ_p\)，其中\(s\in\mathbb{N}_0\)。在此框架下，我们研究非交换二项式公式\((X + Y)^n\)及相关恒等式。特别地，展示了相关正规序系数如何用\((p,q)\) - 变形的\(s\) - 车数表示。我们明确处理了几个特殊情况，得到文献中的已知结果并推导出新结果。
英文摘要
We study the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$ and $Z_p$ satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. Within this framework, we investigate the noncommutative binomial formula $(X+Y)^n$ and related identities. In particular, we show how the associated normal ordering coefficients can be expressed in terms of $(p,q)$-deformed $s$-rook numbers. We treat several special cases explicitly, recovering known results from literature as well as deriving new ones.
Comments36 pages
