Fibonacci-Lucas zeta 和的消去轮廓
Cancellation profiles of Fibonacci-Lucas zeta sums
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中文总结 AI 辅助
研究Fibonacci-Lucas zeta和的项量级分布，证明归一化绝对项和的指数展开及收敛性，并确定最大项索引与条件数增长。
中文摘要 AI 辅助
涉及负整数处 zeta 值的有限 Fibonacci 和与 Lucas 和，其个别项可能远大于总和。本文研究当偶数次数增大时这些项量级的分布。对于至多指数增长的非负权重，我们证明了归一化绝对项和的有限指数展开，并在每个固定阶数处给出显式余项。归一化量级在总变差意义下收敛到一个离散分布，该分布由权重的偶指数生成函数决定。对于 Fibonacci 和与 Lucas 和，该分布在索引八处取得唯一最大值，且我们证明在至少二十六的每个偶数次数下，同一索引给出最大的有限和项。我们还推导了求和条件数的阶乘增长。有符号恒等式被置于已建立的 Bernoulli 多项式反射框架内，并推广到迹一递推族，包括重根和消失尺度。精确有理计算和精化精度评估说明了所证界，并将固定消去轮廓与和的长度的增长区分开来。
英文摘要
Finite Fibonacci and Lucas sums involving zeta values at negative integers can have individual terms far larger than their total. In this paper, we study the distribution of these term magnitudes as the even degree increases. For non-negative weights of at most exponential growth, we prove a finite exponential expansion for the normalized absolute-term sum with an explicit remainder at every fixed order. The normalized magnitudes converge in total variation to a discrete distribution determined by the even exponential generating function of the weights. For the Fibonacci and Lucas sums, this distribution has its unique maximum at index eight and we prove that the same index gives the largest finite-sum term at every even degree at least twenty-six. We also derive the factorial growth of the summation condition number. The signed identities are placed within the established Bernoulli-polynomial reflection framework and extended to a trace-one recurrence family, including repeated roots and a vanishing scale. Exact rational calculations and precision-refined evaluations illustrate the proved bounds and distinguish a fixed cancellation profile from the growing length of the sum.
