Abstract
In this paper, we give a recursive relation for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number, without the Bernoulli numbers and polynomials.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Berndt, B.C.: Ramanujans Notebooks Part V. Springer, Berlin (1998)
Chen, C.P.: Stirling expansions into negative powers of a triangular number. Ramanujan J. 39, 107–116 (2014)
Chen, C.P.: On the coefficients of asymptotic expansion for the harmonic number by Ramanujan. Ramanujan J. 40, 279–290 (2016)
Chen, C.P.: Ramanujan’s formula for the harmonic number. Appl. Math. Comput. 317, 121–128 (2018)
Chen, C.P., Cheng, J.X.: Ramanujan’s asymptotic expansion for the harmonic numbers. Ramanujan J. 38, 123–128 (2015)
Chen, C.P., Li, L.: Two accelerated approximations to the Euler–Mascheroni constant. Sci. Magna 6, 99–107 (2010)
Feng, L., Wang, W.: Riordan array approach to the coefficients of Ramanujans harmonic Number expansion. Results Math. 71, 1413–1419 (2017)
Hirschhorn, M.D.: Ramanujan’s enigmatic formula for the harmonic series. Ramanujan J. 27, 343–347 (2012)
Issaka, A.: An asymptotic series related to Ramanujan’s expansion for the \(n\)th harmonic number. Ramanujan J. 39, 303–313 (2016)
Mortici, C.: On the Ramanujan–Lodge harmonic number expansion. Appl. Math. Comput. 251, 423–430 (2015)
Mortici, C., Chen, C.P.: On the harmonic number expansion by Ramanujan. J. Inequal. Appl. 2013, 222 (2013)
Nemes, G.: Asymptotic expansion for \(\log n!\) in terms of the reciprocal of a triangular number. Acta Math. Hung. 129, 254–262 (2010)
Ramanujan, S.: Notebook II. Narosa, New Delhi (1988)
Villarino, M.B.: Ramanujan’s harmonic number expansion into negative powers of a triangular number. J. Inequal. Pure Appl. Math. 9(3), Article 89 (2008) http://www.emis.de/journals/JIPAM/images/245_07_JIPAM/245_07.pdf
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, CP. On the Ramanujan Harmonic Number Expansion. Results Math 74, 4 (2019). https://doi.org/10.1007/s00025-018-0925-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-018-0925-3