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.

中文翻译：

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微分算子中的多项式和某个收敛级数之和的公式

摘要

令\(P_n(x)\)是任何次数为\(n\geq 2\)且具有实系数的多项式，使得\(P_n(k)\ne 0\)对于\(k\in\mathbb{Z}\ ) . 在论文中，特别地，一系列形式为\(\sum_{k=-\infty}^{+\infty}1/P_n(k)\)的和表示为\((0 ,0)\)由微分表达式\(l_n[y]=P_n(i\,d/dx) y\)和边界条件\(y^{(j )}(0)=y^{(j)}(2\pi)\) ( \(j=0,1,\dots,n-1\)）。因此，这样的和以易于构造的初等函数的值明确表示。显然，这些公式也适用于\(\sum_{k=0}^{+\infty}1/P_n(k^2)\)形式的和，而众所周知，类似的和的通用公式\(\sum_{k=0}^{+\infty}1/P_n(k)\)不存在。