单位根加权三角幂和：常数项方法
Root-of-unity weighted trigonometric power sums: a constant term approach
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中文总结 AI 辅助
该研究提出一种统一方法，结合常数项提取、生成函数与部分分式分解，得到单位根加权的四类三角函数幂和的闭式表达式，可系统重现经典三角恒等式。
中文摘要 AI 辅助
我们提出一种统一方法，用于计算以本原k次单位根加权的余切、正切、余割和正割的有限幂和。该方法依赖于迭代Laurent级数的常数项提取，结合生成函数与部分分式分解。我们得到这四个函数的所有偶次幂和，以及奇次幂余切、正切和的显式闭式表达式，公式以伯努利多项式、欧拉多项式和通用系数r_{n,t}表示。作为应用，我们以系统且初等的方式重现了众多经典恒等式，包括普通和交替余切幂和、以及阿克顿交替正切和。
英文摘要
We present a unified method for evaluating finite sums of powers of cotangent, tangent, cosecant and secant weighted by primitive $k$th roots of unity. The approach relies on constant term extraction for iterated Laurent series, combined with generating functions and partial fraction decomposition. We obtain explicit closed-form expressions for all even-power sums of the four functions, as well as for odd-power cotangent and tangent sums. The formulas are given in terms of Bernoulli polynomials, Euler polynomials, and universal coefficients $r_{n,t}$. As applications, we recover numerous classical identities---including the ordinary and alternating cotangent power sums and Acton's alternating tangent sum---in a systematic and elementary way.
