论文标题
通过持续分数的交替序列的非理性性和超越性
Irrationality and Transcendence of Alternating Series Via Continued Fractions
论文作者
论文摘要
Euler给出了将两种类型的交替系列I和II转换为等效的持续分数的食谱，即其收敛等于部分总和的分数。我们证明了持续分数的非理性性的条件，然后可以简单地证明$ e，\ sin1 $和基础常数是非理性的。我们的主要结果是，如果一系列II类型等同于简单的持续分数，那么总和是超越性的，其非理性性措施超过$ 2 $。我们构造所有$ \ aleph_0^{\ aleph_0} = \ mathfrak {c} $这样的系列，并恢复davison--shallit和cahen常数的超越性。在此过程中，我们提到$π$，黄金比率，费马特，斐波那契和liouville编号，西尔维斯特的序列，皮尔斯扩张，马勒的方法，恩格尔系列，恩格尔系列和兰伯特，Sierpiński和thue-siegel-siegel-siegel-roth。我们还做出了三个猜想。 （该手稿是死后提交的。作者于2020年1月16日去世。）
Euler gave recipes for converting alternating series of two types, I and II, into equivalent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then allows easy proofs that $e,\sin1$, and the primorial constant are irrational. Our main result is that, if a series of type II is equivalent to a simple continued fraction, then the sum is transcendental and its irrationality measure exceeds $2$. We construct all $\aleph_0^{\aleph_0}=\mathfrak{c}$ such series and recover the transcendence of the Davison--Shallit and Cahen constants. Along the way, we mention $π$, the golden ratio, Fermat, Fibonacci, and Liouville numbers, Sylvester's sequence, Pierce expansions, Mahler's method, Engel series, and theorems of Lambert, Sierpiński, and Thue-Siegel-Roth. We also make three conjectures. (This manuscript was submitted posthumously. The author passed away on January 16, 2020.)
