Expansions of generalized Euler's constants into the series of polynomials in and into the formal enveloping series with rational coefficients only

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• 作者：Iaroslav V. Blagouchine¹ ; ^{iaroslav.blagouchine@univ-tln.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Generalized Euler's constants ; Stieltjes constants ; Stirling numbers ; Factorial coefficients ; Series expansion ; Divergent series ; Semi-convergent series ; Formal series ; Enveloping series ; Asymptotic expansions ; Approximations ; Bernoulli numbers ; Harmonic numbers ; Rational coefficients ; Inverse pi
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：158
• 期：Complete
• 页码：365-396
• 全文大小：940 K

文摘

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) 纬_m${纬}_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in 蟺⁻²${蟺}^{-2}$ with rational coefficients and converges slightly better than Euler's series ∑n⁻²$\sum n^{-2}$. The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A, the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them.