论文标题
转移的二项式定理和三角学系列
A shifted binomial theorem and trigonometric series
论文作者
论文摘要
我们介绍了二项式定理的转移版本，并使用它来研究一些显着的三角集积分及其在二项式多数总和方面的明确重写。在各种晶格上的封闭步行计数中产生的区域产生函数的表达式的动机，我们提出了类似的总和，涉及该区域的分数值，并表明它们与整数密切相关，并导致合理序列融合到$π$的幂中。除了数学兴趣之外，我们的结果可能与涉及较高旋转或$ su（n）$自由度的海森堡链类型的统计机械模型的概括有关。
We introduce a shifted version of the binomial theorem, and use it to study some remarkable trigonometric integrals and their explicit rewriting in terms of binomial multiple sums. Motivated by the expressions of area generating functions arising in the counting of closed walks on various lattices, we propose similar sums involving fractional values of the area and show that they are closely related to their integer counterparts and lead to rational sequences converging to powers of $π$. Our results, other than their mathematical interest, could be relevant to generalizations of statistical mechanical models of the Heisenberg chain type involving higher spins or $SU(N)$ degrees of freedom.
