AI 中文总结
该研究证明关联$k$-卢卡斯与$k$-斐波那契序列的二项式恒等式，结合环域原理得到含$n$次复多项式零点的$k$-卢卡斯环域，补充相关基于$k$-斐波那契的界。
AI 中文摘要
我们证明了一个关联$k$-卢卡斯序列与$k$-斐波那契序列的二项式恒等式：对于实数$k>0$、整数$m\geq1$、$n\geq1$，当$n$为偶数时，$k$-卢卡斯数的二项式加权和可简化为$L_{k,mn}$的倍数；当$n$为奇数时，则可简化为$F_{k,mn}$的倍数。这种奇偶性依赖源于$k$-卢卡斯的比内公式缺少$k$-斐波那契数所具有的归一化因子$1/(\alpha-\beta)$。该恒等式针对$n$的每种奇偶性和每个$m$，提供了一族和为1的显式正权重；结合Dalal与Govil的一般环域原理，可得到一个包含所有$n$次复多项式零点的$k$-卢卡斯环域，补充了Díaz-Barrero、Bidkham-Shashahani及Kaur提出的基于$k$-斐波那契的界。
英文摘要
We prove a binomial identity relating the $k$-Lucas and $k$-Fibonacci sequences: for real $k>0$ and integers $m\ge1$, $n\ge1$, a binomial-weighted sum of $k$-Lucas numbers reduces to a multiple of $L_{k,mn}$ when $n$ is even, but to a multiple of $F_{k,mn}$ when $n$ is odd. This parity dependence traces back to the $k$-Lucas Binet formula lacking the normalising factor $1/(α-β)$ present for $k$-Fibonacci numbers. The identity supplies, for each parity of $n$ and each $m$, an explicit family of positive weights summing to one; combined with the general annulus principle of Dalal and Govil, this yields a $k$-Lucas annulus containing all the zeros of a complex polynomial of degree $n$, complementing the $k$-Fibonacci-based bounds of Díaz-Barrero, Bidkham-Shashahani, and Kaur.
CommentsThe k-Lucas sequence is a particular Horadam sequence, and the identity in Theorem 1.1 follows as a specialization of known binomial identities for Horadam sequences, which I had not seen at first
