Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to $π^{-1}$
Iaroslav V. Blagouchine
备注：
The paper was accepted for publication in Mathematics of Computation (AMS) on December 3, 2014. However, due to a conflict with the managing editor of this journal during the production of the paper, I withdrew it and, on September 10, 2015, re-submitted it to the Journal of Mathematical Analysis and Applications (Elsevier)
发表信息：
Journal of Mathematical Analysis and Applications (Elsevier), vol. 442, no. 2, pp. 404-434, 2016
In this paper, two new series for the logarithm of the $Γ$-function are presented and studied. Their polygamma analogs are also obtained and discussed. These series involve the Stirling numbers of the first kind and have the property to contain only rational coefficients for certain arguments related to $π^{-1}$. In particular, for any value of the form $\lnΓ(\frac{1}{2}n\pmαπ^{-1})$ and $Ψ_k(\frac{1}{2}n\pmαπ^{-1})$, where $Ψ_k$ stands for the $k$th polygamma function, $α$ is positive rational greater than $\frac{1}{6}π$, $n$ is integer and $k$ is non-negative integer, these series have rational terms only. In the specified zones of convergence, derived series converge uniformly at the same rate as $\sum(n\ln^m\!n)^{-2}$, where $m=1, 2, 3,\ldots$\,, depending on the order of the polygamma function. Explicit expansions into the series with rational coefficients are given for the most attracting values, such as $\lnΓ(π^{-1})$, $\lnΓ(2π^{-1})$, $\lnΓ(\tfrac{1}{2}+π^{-1})$, $Ψ(π^{-1})$, $Ψ(\tfrac{1}{2}+π^{-1})$ and $Ψ_k(π^{-1})$. Besides, in this article, the reader will also find a number of other series involving Stirling numbers, Gregory's coefficients (logarithmic numbers, also known as Bernoulli numbers of the second kind), Cauchy numbers and generalized Bernoulli numbers. Finally, several estimations and full asymptotics for Gregory's coefficients, for Cauchy numbers, for certain generalized Bernoulli numbers and for certain sums with the Stirling numbers are obtained. In particular, these include sharp bounds for Gregory's coefficients and for the Cauchy numbers of the second kind.
Expansions of generalized Euler's constants into the series of polynomials in $π^{-2}$ and into the formal enveloping series with rational coefficients only
A refined non-vanishing of the $p$-adic logarithm of a rational point on an abelian variety
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Princeton University（普林斯顿大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Étude du cas rationnel de la théorie des formes linéaires de logarithmes. (French) [Study of the rational case of the theory of linear forms in logarithms]
Effective irrationality measures for real and $p$-adic roots of rational numbers close to $1$, with an application to parametric families of Thue-Mahler equations
