A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations

JOURNAL OF NUMBER THEORY

卷:148

Article

Blagouchine, Iaroslav V.

关键词: Stieltjes constants;    Generalized Euler's constants;    Special constants;    Number theory;    Zeta function;    Gamma function;    Digamma function;    Psi function;    Malmsten;    Rational arguments;    Logarithmic integrals;    Logarithmic series;    Complex analysis;    Orthogonal expansions;   

DOI  :  10.1016/j.jnt.2014.08.009

来源: Elsevier

【 摘 要 】

Recently, it was conjectured that the first generalized Stieltjes constant at rational argument may be always expressed by means of Euler's constant, the first Stieltjes constant, the Gamma-function at rational argument(s) and some relatively simple, perhaps even elementary, function. This conjecture was based on the evaluation of gamma(1)(1/2), gamma(1)(1/3), gamma(1)(2/3), gamma(1)(1/4), gamma(1)(3/4), gamma(1)(1/6), gamma(1)(5/6), which could be expressed in this way. This article completes this previous study and provides an elegant theorem which allows to evaluate the first generalized Stieltjes constant at any rational argument. Several related summation formulae involving the first generalized Stieltjes constant and the Digamma function are also presented. In passing, an interesting integral representation for the logarithm of the Gamma-function at rational argument is also obtained. Finally, it is shown that similar theorems may be derived for higher Stieltjes constants as well; in particular, for the second Stieltjes constant the theorem is provided in an explicit form. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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