在半平面等分角上正割与余割函数幂的乘积和

Summations the Products of the Powers of Secant and Cosecant Functions on the Upper Half-plane

研究含有一个自由参数在半平面等分角上的正割与余割函数幂的乘积求和问题.使用双变量发生函数的方法和部分分式分解技巧,按照半平面等分次数的奇偶性分别计算有限和与交错和,给出了4种情况下的显式求和公式,也列出了一些低次幂乘积和的简洁结果.

The product sum of the powers of secant and cosecant functions at the equivalent angles of half-plane with one free parameter is investigated.Using the method of bivariate generating function and partial fractional decomposition technique,both the finite sum and alternating sum are calculated according to the parity of the half-plane equivalent number.The explicit summation formulas in four cases are given,and some simple results for the product sum of lower powers are also listed.