Polynomials in the differentiation operator and formulas for the sums of some converging series

Let $P_n(x)$ be any polynomial of degree $n\geq 2$ with real coefficients such that $P_n(k)\ne 0$ for $k\in\mathbb{Z}$. In the paper, in particular, the sum of a series of the form $\sum_{k=-\infty}^{+\infty}1/P_n(k)$ is expressed as the value at $(0,0)$ of the Green function of the self-adjoint problem generated by the differential expression $l_n[y]=P_n(i\,d/dx) y$ and the boundary conditions $y^{(j)}(0)=y^{(j)}(2\pi)$ ($j=0,1,\dots,n-1$). Thus, such a sum is explicitly expressed in terms of the value of an easy-to-construct elementary function. These formulas, obviously, also apply to sums of the form $\sum_{k=0}^{+\infty}1/P_n(k^2)$, while it is well known that similar general formulas for the sum $\sum_{k=0}^{+\infty}1/P_n(k)$ do not exist.