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 XVIIIth century.

翻译：本文研究余割的有限和 $\sum\csc\big(\varphi+a\pi l/n\big)$，其中指标 $l$ 从 1 取至 $n-1$，$\varphi$ 和 $a$ 为任意参数，以及若干密切相关的和，如正割、正切和余切的类似和。这些三角和在数学、物理学及多种相关学科的各种问题中出现。其特定情形在先前工作中被零星考虑过，且注意到即使简单特例 $\sum\csc\big(\pi l/n\big)$ 也不存在闭式，即紧凑的求和公式。在本文中，我们推导上述和的几种替代表示，研究它们的性质，将其与许多其他有限和及无限和关联，获得它们关于大 $n$ 的完整渐近展开，并提供精确的上下界（例如，对于 $n\geqslant10$，上界的典型相对误差小于 $2\times10^{-9}$；对于 $n\geqslant50$，小于 $7\times10^{-14}$，这远优于我们能在先前工作中找到的界）。我们的研究表明这些和与若干特殊数和函数密切相关，尤其是双伽马函数（此外，作为副产品，我们获得双伽马函数的几个有趣求和公式）。渐近研究显示，这些和可能因 $\varphi$ 和 $a$ 的选择而呈现定性不同的行为；特别地，随着 $n$ 增大，其中一些可能偶发性地变大。最后，我们提供与本文所考虑各种和相关的若干历史评注。我们表明，该领域的某些结果要么被多次重新发现，要么可以轻易从各种已知公式（包括一些可追溯至十八世纪的公式）推导得出。