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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 kernelassociated to the combinatorial Laplacian which acts on the space of functionson $X_{m}$ 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 spectralexpansion of the Laplacian, we show that $G_{X_m,\chi_{\beta}}(x,y;s)$ is agenerating function for certain trigonometric sums involving powers of thecosecant function; by choosing $\beta$ or $s$ appropriately, the sums inquestion involve powers of the secant function. Second, by viewing $X_{m}$ as aquotient space of $\mathbb{Z}$, we prove that $G_{X_m,\chi_{\beta}}(x,y;s)$ isa rational function which is given in terms of Chebyshev polynomials. From theexistence and uniqueness of $G_{X_m,\chi_{\beta}}(x,y;s)$, these twoevaluations are equal. From the resulting identity, we obtain a means by whichone can obtain explicit evaluations of cosecant and secant sums. The identitieswe prove depend on a number of parameters, and when we specialize the values ofthese parameters we obtain several previously known formulas. Going further, wederive a recursion formula for special values of the $L$-functions associatedto the cycle graph $X_{m}$, thus answering a question from arXiv:2212.13687v1.

The resolvent kernel on the discrete circle and twisted cosecant sums