On a generalization of Watson's trigonometric sum, some of its properties and its relationship to Dowker's sum

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.