Which Partial Sums of the Taylor Series for $e$ are Convergents to $e$? (and a Link to the Primes 2, 5, 13, 37, 463), II
Jonathan Sondow, Kyle Schalm
备注：
16 pages, 6 tables. In the new Section 4, we prove that if p/q is a convergent to e with q = (n!)^k for some n and k > 0, then n/k < e. The old Section 4 is now Section 5. Several parts of the Appendix are clarified. To appear in ``Tapas in Experimental Mathematics 2009'' in the AMS series Contemporary Mathematics
发表信息：
"Gems in Experimental Mathematics" (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010
This is an expanded version of our earlier paper. Let the $n$th partial sum of the Taylor series $e = \sum_{r=0}^{\infty} 1/r!$ be $A_n/n!$, and let $p_k/q_k$ be the $k$th convergent of the simple continued fraction for $e$. Using a recent measure of irrationality for $e$, we prove weak versions of our conjecture that only two of the partial sums are convergents to $e$. A related result about the denominators $q_k$ and powers of factorials is proved. We also show a surprising connection between the $A_n$ and the primes 2, 5, 13, 37, 463. In the Appendix, we give a conditional proof of the conjecture, assuming a second conjecture we make about the zeros of $A_n$ and $q_k$ modulo powers of 2. Tables supporting this Zeros Conjecture are presented and we discuss a 2-adic reformulation of it.
Intermediate convergents and a metric theorem of Khinchin
连分数度量理论中的一个里程碑式定理是这样开始的：选择一个定义在正整数上的非负实函数$f$和一个实数$x$，并形成$f$在$x$的连分数展开式中的偏导数$a_1,..., a_n$处求值的部分和$s_n$。序列$\{s_n/n\}$是否有与$n\rar\infty$相同的限值？1935年A. Y. Khinchin证明了，只要函数$f$增长得不太快，几乎所有$x$的答案都是肯定的。在本文中，我们将探讨这个问题的一个自然的重新制定，其中的功能$f$定义在有理数和部分和的问题是在中间收敛到$x$，具有小于规定数量的收敛因子。通过使用一些欣钦的想法连同更现代的结果，我们能够提供一个定量渐近定理类似于上述经典之一。
本文研究了无穷收敛和$T=\sum_{n=0}^\infty\frac{P(n)}{Q(n)}$，其中$P(x)\in\bar{\mathbb{Q}}[x]$，$Q(x)\in\mathbb{Q}[x]$和$Q(x)$只有单有理零点. N. Saradha和R. Tijdeman等得到了当$Q(x)$的次数为3时$T$超越的充必要条件。本文给出了当$Q(x)$的度为4且$Q(x)$是约化的时，$T$超越的充必要条件.
Taylor's series expansions for real powers of functions containing squares of inverse (hyperbolic) cosine functions, explicit formulas for special partial Bell polynomials, and series representations for powers of circular constant
