AI 中文总结
研究通过分母函数求有理指数的斐波那契数和卢卡斯数的逆，推导\(F_X\)的一般显式公式，确定其与卢卡斯数及斐波那契序列的关系，计算相关数的单位，证明乘法逆元存在，揭示序列新结构性质。
AI 中文摘要
斐波那契数和卢卡斯数在各种代数和数论背景下都有广泛研究，包括它们的模逆和推广。受这些进展的启发，我们通过分母函数\(F\)研究有理指数的斐波那契数和卢卡斯数的逆。Uludağ和Gökmen（2022）表明，有理指数的斐波那契数\(F_X\)（\(X\in\mathbb{Q}_{>0}\)）可用\(F\)表示且有无限多种表示。本文通过分母函数为\(F_X\)推导一般显式公式扩展了他们的工作。我们确定了\(F_X\)与卢卡斯数一致或偏离经典斐波那契序列的精确条件。此外，借助这些广义公式，我们计算了斐波那契数和卢卡斯数的单位，从而证明在此框架内所有斐波那契数和卢卡斯数都存在乘法逆元。这些结果揭示了与斐波那契和卢卡斯相关序列的新结构性质，并为数论探索指明了进一步方向。
英文摘要
Fibonacci and Lucas numbers have been extensively studied in various algebraic and number-theoretic contexts, including their modular inverses and generalizations. Motivated by these developments, we study the inverses of Fibonacci and Lucas numbers with rational indices through the codenominator function $F$. Uludağ and Gökmen (2022) showed that the rational-indexed Fibonacci number $F_X$, where $X\in\mathbb{Q}_{>0}$, can be expressed using $F$, and that infinitely many such representations exist. In this paper, we extend their work by deriving a general explicit formula for $F_X$ through the codenominator function. We establish precise conditions under which $F_X$ coincides with a Lucas number or deviates from the classical Fibonacci sequence. Moreover, by means of these generalized formulas, we compute the units of Fibonacci and Lucas numbers, thereby proving the existence of multiplicative inverses for all Fibonacci and Lucas numbers within this framework. These results reveal new structural properties of Fibonacci- and Lucas-related sequences and suggest further directions for number-theoretic exploration.
Comments14 pages. Final version. To appear in the Fibonacci Quarterly
DOI：10.1080/00150517.2026.2638782
