Abstract
We study a sequence of constants known as the Bendersky-Adamchik constants which appear quite naturally in number theory and generalize the classical Glaisher-Kinkelin constant. Our main initial purpose is to elucidate the close relation between the logarithm of these constants and the Ramanujan summation of certain divergent series. In addition, we also present a remarkable, and previously unknown, expansion of the logarithm of these constants in convergent series involving the Bernoulli numbers of the second kind.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availibility
Data sharing not applicable to this article as no datasets were generated or analysed during the current study
Notes
According to Kellner [1, Rem. 27] this expression of \(\ln \Gamma _k\) is due to Alexeiewsky in the special case where \(x=n+1\) is an integer.
References
Kellner, B.C.: On asymptotic constants related to products of Bernoulli numbers. Integers 9, 83–106 (2009)
Bendersky, L.: Sur la function gamma généralisée. Acta Math. 61, 263–322 (1933)
Sondow, J., Hadjicostas, P.: The generalized-Euler-constant function \(\gamma (z)\) and a generalization of Somos’s quadratic recurrence constant. J. Math. Anal. Appl. 332, 292–314 (2007)
Kurokawa, N., Ochiai, H.: Generalized Kinkelin’s formulas. Kodai Math. J. 30, 195–212 (2007)
Cohen, H.: Number Theory, , Volume II: Analytic and Modern Tools, Graduate Texts in Math, vol. 240. Springer, Germany (2007)
Candelpergher, B.: Ramanujan Summation of Divergent Series, Lecture Notes in Math. 2185. Springer, Germany (2017)
Wang, W.: Some asymptotic expansions of hyperfactorial functions and generalized Glaisher-Kinkelin constants. Ramanujan J. 43, 513–533 (2017)
Adamchik, V.: Polygamma functions of negative order. J. Comput. Appl. Math. 100, 191–199 (1998)
Perkins, M., Van Gorder, R.A.: Closed-form calculation of infinite products of Glaisher-type related to Dirichlet series. Ramanujan J. 49, 371–389 (2019)
Candelpergher, B., Coppo, M.-A.: A new class of identities involving Cauchy numbers, harmonic numbers and zeta values. Ramanujan J. 27, 305–328 (2012)
Coppo, M.A., Young, P.T.: On shifted Mascheroni series and hyperharmonic numbers. J. Number Theory 169, 1–20 (2016)
Blagouchine, I.V.: A complement to a recent paper on some infinite sums with the zeta values, preprint, 2020. Available at arxiv:2001.00108
Coppo, M.-A.: A note on some alternating series involving zeta and multiple zeta values. J. Math. Anal. Appl. 475, 1831–1841 (2019)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
conflicts of interest
The author has no conflicts of interest to disclose
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Coppo, MA. Generalized Glaisher-Kinkelin constants and Ramanujan summation of series. Res. number theory 10, 15 (2024). https://doi.org/10.1007/s40993-023-00505-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-023-00505-2