Polynomials in the Differentiation Operator and Formulas for the Sums of Certain Convergent Series

Abstract

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.