二阶线性递推倒数级数中的阿贝尔型变换与伸缩结构
Abel-Type Transformations and Telescoping Structures in Reciprocal Series of Second-Order Linear Recurrences
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中文总结 AI 辅助
研究二阶线性递推倒数级数变换与求值，基于离散阿贝尔型求和公式，将含多因子的倒数级数化为部分伸缩结构，得到相关变换公式，统一扩展已知恒等式，还能处理经典组合序列，有效简化多因子倒数和。
中文摘要 AI 辅助
我们开发了一种统一的方法来变换和评估分母中包含二阶线性递推项乘积的无穷级数。该方法基于离散阿贝尔型求和公式（分部求和），将具有三个或更多因子的倒数级数转换为呈现部分伸缩结构的表达式。由此，我们得到了形如\(\sum\limits_{k=1}^{\infty} \frac{(\pm 1)^k}{w_{rk+l} w_{mk+s} w_{m(k+1)+s}}\)的级数的一般变换公式，以及对四个或更多项乘积的扩展。这些公式提供了一个统一和扩展许多斐波那契数和卢卡斯数已知恒等式的系统框架。此外，该方法还能得到涉及几个经典组合序列的显式求值和恒等式。该方法的一个关键特性是它能自然地区分参数\(m\)的偶数和奇数取值，从而得到结构不同的表示。结果表明，分部求和是将多因子倒数和简化为更简单形式的有效且灵活的工具。
英文摘要
We develop a unified method for transforming and evaluating infinite series involving products of terms of second-order linear recurrences in the denominator. The approach is based on a discrete Abel-type summation formula (summation by parts), which converts reciprocal series with three or more factors into expressions exhibiting a partial telescoping structure. As a consequence, we obtain general transformation formulas for series of the form $\sum\limits_{k=1}^{\infty} \frac{(\pm 1)^k}{w_{rk+l} w_{mk+s} w_{m(k+1)+s}}$, together with extensions to products of four or more terms. These formulas provide a systematic framework that unifies and extends many known identities for Fibonacci and Lucas numbers. In addition, the method leads to explicit evaluations and identities involving several classical combinatorial sequences, including Catalan numbers, harmonic numbers, and Stirling numbers of both kinds. A key feature of the approach is that it naturally distinguishes between even and odd values of the parameter $m$, leading to structurally different representations. The results show that summation by parts is an effective and flexible tool for reducing multi-factor reciprocal sums to simpler forms.
