AI 中文总结
本文推广Zeckendorf定理至二阶线性递推，分析展开计数渐近行为，给出唯一展开充分条件，并完全刻画g-黄金比例递推族。
AI 中文摘要
我们将Zeckendorf定理推广到形如$G_{k+2} = gG_{k+1} + hG_{k}$的二阶线性递推，其中初始值$(G_1, G_2)$为任意互素的整数。我们分析了在递推关联的标准展开规则$\mathcal{E}$下，小于或等于$X$且具有展开的正整数个数$\\#R_G(X)$的渐近行为。此外，我们给出了$G$在展开规则下具有唯一展开性质的充分条件。最后，当$G_1 = 1$或$G_2 = 1$时，我们完全刻画了$g$-黄金比例递推族。我们的方法直接研究展开的代数结构，简化并完善了先前的部分结果。
英文摘要
We generalize Zeckendorf's theorem to second-order linear recurrences of the form $G_{k+2} = gG_{k+1} + hG_{k}$ with arbitrary, coprime initial values $(G_1, G_2)$. We analyze the asymptotic behavior of the count $\#R_G(X)$ of positive integers less than or equal to $X$ that have an expansion under the standard rule of expansions $\mathcal{E}$ associated with the recurrence. Additionally, we provide sufficient conditions under which $G$ has the unique expansion property under the rule of expansions. Finally, we completely characterize the $g$-golden ratio recurrence family when $G_1 = 1$ or $G_2 = 1$. Our approach directly investigates the algebraic structure of the expansions, simplifying and completing previous partial results.
Comments44 pages
