In this paper we investigate the finite sum of cosecants $\sum\csc\big(\varphi+a\pi l/n\big),$ where the index $l$ runs through 1 to $n-1$ and $\varphi$ and $a$ are arbitrary parameters, as well as several closely related sums, such as similar sums of a series of secants, of tangents and of cotangents. These trigonometric sums appear in various problems in mathematics, physics, and a variety of related disciplines. Their particular cases were fragmentarily considered in previous works, and it was noted that even a simple particular case $\sum\csc\big(\pi l/n\big)$ does not have a closed-form, i.e.~a compact summation formula. In the paper, we derive several alternative representations for the above-mentioned sums, study their properties, relate them to many other finite and infinite sums, obtain their complete asymptotic expansions for large $n$ and provide accurate upper and lower bounds (e.g. the typical relative error for the upper bound is lesser than $2\times10^{-9}$ for $n\geqslant10$ and lesser than $7\times10^{-14}$ for $n\geqslant50$, which is much better than the bounds we could find in previous works). Our researches reveal that these sums are deeply related to several special numbers and functions, especially to the digamma function (furthermore, as a by-product, we obtain several interesting summations formulae for the digamma function). Asymptotical studies show that these sums may have qualitatively different behaviour depending on the choice of $\varphi$ and $a$; in particular, as $n$ increases some of them may become sporadically large. Finally, we also provide several historical remarks related to various sums considered in the paper. We show that some results in the field either were rediscovered several times or can easily be deduced from various known formulae, including some formulae dating back to the XIIXth century.

翻译：本文研究余割有限和 $\sum\csc\big(\varphi+a\pi l/n\big)$，其中指标 $l$ 从1取至 $n-1$，$\varphi$ 和 $a$ 为任意参数，同时研究数个密切相关之和，例如正割、正切和余切的类似和。这些三角和出现在数学、物理学及众多相关学科的各种问题中。其特例在先前工作中被零星讨论，且已注意到即使简单特例 $\sum\csc\big(\pi l/n\big)$ 也不存在闭形式，即紧凑求和公式。本文推导了上述和的若干替代表示，研究其性质，将其与许多其他有限及无限和建立关联，获得其当 $n$ 较大时的完整渐近展开，并提供精确上下界（例如，当 $n\geqslant10$ 时上界典型相对误差小于$2\times10^{-9}$，当 $n\geqslant50$ 时小于$7\times10^{-14}$，远超先前工作中的界）。我们的研究表明，这些和与若干特殊数和函数（尤其是双伽马函数）深度关联；此外，作为副产品，我们获得了双伽马函数的若干有趣求和公式。渐近分析表明，根据 $\varphi$ 和 $a$ 的选取，这些和可能呈现定性不同的行为；特别是，随着 $n$ 增大，其中某些和可能偶尔变得极大。最后，我们还提供与文中各类和相关的若干历史注记，表明该领域部分结果或被多次重新发现，或可由已知公式（包括可追溯至十八世纪的某些公式）直接推导得出。