The resolvent kernel on the discrete circle and twisted cosecant sums

Let $X_m$ denote the discrete circle with $m$ vertices. For $x,y\in X_{m}$ and complex $s$, let $G_{X_m,\chi_{\beta}}(x,y;s)$ be the resolvent kernel associated to the combinatorial Laplacian which acts on the space of functions on $X_{m}$ that are twisted by a character $\chi_{\beta}$. We will compute $G_{X_m,\chi_{\beta}}(x,y;s)$ in two different ways. First, using the spectral expansion of the Laplacian, we show that $G_{X_m,\chi_{\beta}}(x,y;s)$ is a generating function for certain trigonometric sums involving powers of the cosecant function; by choosing $\beta$ or $s$ appropriately, the sums in question involve powers of the secant function. Second, by viewing $X_{m}$ as a quotient space of $\mathbb{Z}$, we prove that $G_{X_m,\chi_{\beta}}(x,y;s)$ is a rational function which is given in terms of Chebyshev polynomials. From the existence and uniqueness of $G_{X_m,\chi_{\beta}}(x,y;s)$, these two evaluations are equal. From the resulting identity, we obtain a means by which one can obtain explicit evaluations of cosecant and secant sums. The identities we prove depend on a number of parameters, and when we specialize the values of these parameters we obtain several previously known formulas. Going further, we derive a recursion formula for special values of the $L$-functions associated to the cycle graph $X_{m}$, thus answering a question from arXiv:2212.13687v1.