In this paper we study two finite trigonometric sums $\sum a_l\csc\big(\pi l/n\big)\,,$ where $a_l$ are equal either to $\cos(2\pi l \nu/n)$ or to $(-1)^{l+1}$, and where the summation index $l$ and the discrete parameter $\nu$ both run through $1$ to $n-1$ (marginally, cases $a_l=\ln\csc\big(\pi l/n\big)$ and $a_l=\Psi\big(l/n\big)$ also appear in the paper). These sums occur in various problems in mathematics, physics and engineering, and play an important part in some number-theoretic problems. Formally, the first of these sums is also the so-called Dowker sum of order one half. In the paper, we obtain several integral and series representations for the above-mentioned sums, investigate their properties, derive their complete asymptotical expansions and deduce very accurate upper and lower bounds for them (both bounds are asymptotically vanishing). In addition, we obtain a useful approximate formula containing only three terms, which is also very accurate and can be particularly appreciated in applications. Both trigonometric sums appear to be closely related with the digamma function and with the square of the Bernoulli numbers. Finally, we also derive several advanced summation formulae for the gamma and the digamma functions, in which the first on these sums, as well as the product of a sequence of cosecants $\,\prod\big(\csc(\pi l/n)\big)^{\csc(\pi l/n)}$, play an important role.

翻译：本文研究两个有限三角和 $\sum a_l\csc\big(\pi l/n\big)\,,$ 其中 $a_l$ 取值为 $\cos(2\pi l \nu/n)$ 或 $(-1)^{l+1}$，求和指标 $l$ 与离散参数 $\nu$ 均遍历 $1$ 至 $n-1$（文中亦涉及 $a_l=\ln\csc\big(\pi l/n\big)$ 与 $a_l=\Psi\big(l/n\big)$ 的边际情形）。这些和在数学、物理学及工程学的各类问题中出现，并在某些数论问题中扮演重要角色。形式上，第一个和亦为所谓半阶Dowker和。本文获得了上述和的若干积分与级数表示，研究了其性质，推导了其完整渐近展开式，并给出了非常精确的上下界估计（两者均渐近趋于零）。此外，我们得到了一个仅含三项的有用近似公式，该公式同样具有高精度，在应用中尤具价值。这两个三角和均与双伽玛函数及伯努利数的平方密切相关。最后，我们还推导了伽玛函数与双伽玛函数的若干高级求和公式，其中第一个和以及余割序列的乘积 $\,\prod\big(\csc(\pi l/n)\big)^{\csc(\pi l/n)}$ 起着关键作用。