微分算子中的多项式及收敛级数和的公式

IF 0.6 4区 数学 Q3 MATHEMATICS
K. A. Mirzoev, T. A. Safonova
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引用次数: 1

摘要

设\(P_n(x)\)为任意次为\(n\geq 2\)的多项式,其系数为实数,如\(P_n(k)\ne 0\)为\(k\in\mathbb{Z}\)。特别地,本文将形式为\(\sum_{k=-\infty}^{+\infty}1/P_n(k)\)的一系列的和表示为微分表达式\(l_n[y]=P_n(i\,d/dx) y\)和边界条件\(y^{(j)}(0)=y^{(j)}(2\pi)\) (\(j=0,1,\dots,n-1\))生成的自伴随问题的Green函数在\((0,0)\)处的值。因此,这样的和是用一个易于构造的初等函数的值显式表示的。显然,这些公式也适用于\(\sum_{k=0}^{+\infty}1/P_n(k^2)\)这种形式的和,而众所周知,类似的和的一般公式\(\sum_{k=0}^{+\infty}1/P_n(k)\)并不存在。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Polynomials in the Differentiation Operator and Formulas for the Sums of Certain Convergent 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.

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来源期刊
CiteScore
0.90
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.
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