Closed-form summations of Dowker's and related trigonometric sums (Q2920323)

On a physical base, J. S. Dowker evaluated the (nowadays called) Dowker sums NEWLINE\[NEWLINEC_{2n}(q,r)=\sum_{p=1}^{q-1}\cos\left(\frac{2rp\pi}{q}\right)\csc^{2n}\left(\frac{p\pi}{q}\right).NEWLINE\]NEWLINE Here the authors give a closed form of twelve related sums. These expressions involve Bernoulli polynomials.NEWLINENEWLINEJust to mention one of the results, we cite the following formula: NEWLINE\[NEWLINE\sum_{p=1}^{q-1}\cos\left(\frac{2rp\pi}{q}\right)\cot^{2n}\left(\frac{p\pi}{q}\right)=NEWLINE\]NEWLINE NEWLINE\[NEWLINE\frac{(-1)^{n-1}}{(2n)!}\sum_{\alpha=0}^n\sum_{\beta=0}^{2n}\binom{2n}{2\alpha}\binom{2n}{\beta}B_{2\alpha}\left(\frac{r}{q}\right)B_{2n-2\alpha}^{(2n)}(\beta)q^{2\alpha}.NEWLINE\]NEWLINE Here \(n\in\mathbb N\), \(q\in\mathbb N\setminus\{1\}\) and \(r=1,2,\dots,q-1\). \(B_n^{(m)}(x)\) are the Bernoulli polynomials of order \(m\) and degree \(n\).