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首页> 外文期刊>Journal of Mathematical Analysis and Applications >Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to pi(-1)
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Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to pi(-1)

机译：伽马函数对数的两个级数展开，涉及斯特林数，仅包含与pi（-1）相关的某些自变量的有理系数

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摘要

In this paper, two new series for the logarithm of the P-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 pi(-1). In particular, for any value of the form In Gamma(1/2n +/- alpha pi(-1)) and Psi k(1/2n +/- alpha pi(-1)), where Psi(k) stands for the kth polygamma function, a is positive rational greater than 1/6 pi, 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 Sigma(n In(m)n)(-2), where m = 1, 2, 3, ... , 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 In Gamma(pi(-1)), In Gamma(2 pi(-1)), ln Gamma(1/2 + pi(-1)), Psi(pi(-1)), Psi(1/2 + pi(-1)) and Psi(k)(pi(-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. (C) 2016 Elsevier Inc. All rights reserved.
机译：本文提出并研究了两个新的P函数对数级数。还获得并讨论了它们的聚γ类似物。这些级数涉及第一类斯特林数，并且具有只包含与pi（-1）相关的某些自变量的有理系数的性质。特别是对于形式为In Gamma（1 / 2n +/- alpha pi（-1））和Psi k（1 / 2n +/- alpha pi（-1））的任何值，其中Psi（k）表示第k个多伽马函数，a是大于1/6 pi的正有理数，n是整数，k是非负整数，这些级数只有有理项。在指定的收敛区域中，导出的级数以与Sigma（n In（m）n）（-2）相同的速率均匀收敛，其中m = 1，2，3，...，这取决于多伽玛的阶数功能。对于最吸引人的值，例如In Gamma（pi（-1）），In Gamma（2 pi（-1）），ln Gamma（1/2 + pi（-1），给出了具有有理系数的级数的显式展开。 ）），Psi（pi（-1）），Psi（1/2 + pi（-1））和Psi（k）（pi（-1））。此外，在本文中，读者还将发现许多其他系列，包括斯特林数，格雷戈里系数（对数，也称为第二类伯努利数），柯西数和广义伯努利数。最后，获得了对格雷戈里系数，柯西数，某些广义伯努利数以及某些与斯特林数之和的估计和完全渐近性。特别地，这些包括格雷戈里系数和第二种柯西数的尖锐边界。 （C）2016 Elsevier Inc.保留所有权利。

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