一组代数恒等式及其应用

A Family of Algebraic Identities with Their Applications

设{x_i}为任一复数序列。从两个初等代数恒等式的证明出发,研究了一类求和运算的封闭形式.基本定理可叙述如下;记{θ_j}p 为 p 次单位根,则有■通过对序列{x_j)和变量 t ,τ的特殊选择,上述定理给出一系列关于二项式系数及 Gauss 二项式系数的求和公式。其中包括徐利治、欧阳植(1984～1985)的新近工作作为特款。此外,定理的极限形式还可给出 Euler 关于自然数例数偶次幂和公式的一种新的推导。

With start of proving an algebraic identity this paper investigates the closed forms of some discrete summations.Based on the splitting technique on summations,several algebraic and combinatorial identities are established.As the interesting corollaries,the famous Euierian formulae are demonstrated consequently,i.e.,