论文标题
斐波那契和卢卡斯数的先验系列
Transcendental Series of Reciprocals of Fibonacci and Lucas Numbers
论文作者
论文摘要
令$ f_1 = 1，f_2 = 1，\ ldots $为fibonacci序列。由身份$ \ displayStyle \ sum_ {k = 0}^{\ infty} \ frac {1} {f_ {2^k}} = \ frac {7- \ sqrt {5}} {2}} {2} {2} $，Erdös和Graham激励$ \ displayStyle \ sum_ {k = 1}^{\ infty} \ frac {1} {f_ {n_k}} $对于任何正整数$ n_1，n_2，n_2，\ ldots $带有$ \ frac {n_ frac {n_ frac {k+1 Quq n_n_ $ n_1，n_1，n_2，\ ldots $都是不合理的。我们解决了他们问题的超越对应物：作为我们主要定理的特殊情况，我们有$ \ displayStyle \ sum_ {k = 1}^{\ infty} \ frac {1} {f_ {f_ {n_k}} $是$ \ frac geq { C> 2 $。由于一开始的身份，绑定的$ c> 2 $是最好的。本文提供了一种应用子空间定理以获得超越结果的新方法，并扩展了仅通过Mahler的方法来获得以前的非平凡结果，以获得$ n_k = d^k+r $的特殊序列的特殊序列。
Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erdös and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is irrational for any sequence of positive integers $n_1,n_2,\ldots$ with $\frac{n_{k+1}}{n_k}\geq c>1$. We resolve the transcendence counterpart of their question: as a special case of our main theorem, we have that $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is transcendental when $\frac{n_{k+1}}{n_k}\geq c>2$. The bound $c>2$ is best possible thanks to the identity at the beginning. This paper provides a new way to apply the Subspace Theorem to obtain transcendence results and extends previous non-trivial results obtainable by only Mahler's method for special sequences of the form $n_k=d^k+r$.
