Abstract
Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tables, 9th printing, Applied Mathematics Series,vol.55. Nation Bureau of Standards, Dover New York (1972)
Allasia G., Giordano C., Pečarić J.: Inequalities for the Gamma function relating to asymptotic expansions, Math. Math Inequal. Appl. 5, 543–555 (2002)
Berndt B.C.: Ramanujan’s Notebooks, Part V. Springer, Berlin (1998)
Burić T., Elezović N.: Approximants of the Euler–Mascheroni constant and harmonic numbers. Appl. Math. Comput. 222, 604–611 (2013)
Chen C.-P.: On the coefficients of asymptotic expansion for the harnomic number by Ramanujan. Ramanujan J. 40, 279–290 (2016)
Chen C.-P.: Stirling expansions into negative powers of a triangular number. Ramanujan J. 39, 107–116 (2016)
Chen C.-P., Cheng J.-X.: Ramanujan’s asymptotic expansion for the harmonic numbers. Ramanujan J. 38, 123–128 (2015)
Comtet L.: Advanced Combinatorics, the Art of Finite and Infinite Expansions, D. Reidel Publishing Co., Dordrecht (1974)
Feng L., Wang W.P.: Riordan array approach to the coefficients of Ramanujan’s harmonic number expansion. Results Math. 71, 1413–1419 (2017)
H. W. Gould, Combinatorial Identities, A standardized set of tables listing 500 binomial coefficients summa, Henry W. Gould (Morgantown, W.Va., 1972).
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 nth harmonic number. Ramanujan J. 39, 303–313 (2016)
Mortici C., Chen C.-P.: On the harmonic number expansion by Ramanujan, J. Inequal.Appl. 2013, 222 (2013)
Mortici C., Villarino M.B.: On the Ramanujan–Lodge harmonic number expansion. Appl. Math. Comput. 251, 423–430 (2015)
Nemes G.: Asymptotic expansion for log n! in terms of the reciprocal of a triangular number. Acta Math Hungar. 129, 254–262 (2010)
Ramanujan S.: Notebook II. Narosa, New Delhi (1988)
M. B. Villarino, Ramanujan’s harmonic number expansion into negative powers of a triangular number, J. Inequal. Pure Appl. Math., 9 (2008), Article 89.
Wilf H.S.: Generating Functionology. Academic Press, New York (1990)
Xu A.M.: Ramanujan’s harmonic number expansion and two identities for Bernoulli numbers. Results Math. 72, 1857–1864 (2017)
Acknowledgement
I am grateful to the anonymous referee for careful reading and helpful suggestions on improving the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LY18A010001), the National Natural Science Foundation of China (Grant No. 11201430) and the Ningbo Natural Science Foundation (Grant No. 2017A610140).
Rights and permissions
About this article
Cite this article
Xu, A. Note on the coefficients of Ramanujan’s expansion for the logarithm of the factorial. Acta Math. Hungar. 156, 255–262 (2018). https://doi.org/10.1007/s10474-018-0849-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-018-0849-0