Arabian Journal of Mathematics

, Volume 6, Issue 2, pp 95–108 | Cite as

Expansions of the exponential and the logarithm of power series and applications

Open Access


In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.

Mathematics Subject Classification

34E05 11B73 11B75 11B83 30B10 


  1. 1.
    Chen, C.-P.; Elezović, N.; Vukšić, L.: Asymptotic formulae associated with the Wallis power function and digamma function. J. Class. Anal. 2(2), 151–166 (2013). Available online at doi: 10.7153/jca-02-13
  2. 2.
    Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions, revised and enlarged edn. D. Reidel Publishing Co., Dordrecht (1974)Google Scholar
  3. 3.
    Gould, H.W.: Coefficient identities for powers of Taylor and Dirichlet series. Am. Math. Mon. 81(1), 3–14 (1974). Available online at doi: 10.2307/2318904
  4. 4.
    Graham, R.L.; Knuth, D.E.; Patashnik, O.: Concrete Mathematics—A Foundation for Computer Science, 2nd edn. Addison-Wesley Publishing Company, Reading (1994)Google Scholar
  5. 5.
    Guo, B.-N.; Qi, F.: An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions. Glob. J. Math. Anal. 2(4), 243–248 (2014). Available online at doi: 10.14419/gjma.v2i4.3310
  6. 6.
    Knopp, K.: Theory and Application of Infinite Series, 2nd English edn. Blackie & Son Limited, Glasgow (1951) [Translated from the fourth German edition by Miss R. C. H. Young]Google Scholar
  7. 7.
    Qi, F.: An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers. Mediterr. J. Math. 13(5), 2795–2800 (2016). Available online at doi: 10.1007/s00009-015-0655-7
  8. 8.
    Qi, F.: Some inequalities for the Bell numbers. ResearchGate Technical Report (2015). Available online at doi: 10.13140/RG.2.1.2544.2721 [Proc. Indian Acad. Sci. Math. Sci. 126(4) (2016, in press)]
  9. 9.
    Qi, F.; Mortici, C.: Some inequalities for the trigamma function in terms of the digamma function. Appl. Math. Comput. 271, 502–511 (2015). Available online at doi: 10.1016/j.amc.2015.09.039
  10. 10.
    Qi, F.; Shi, X.-T.; Liu, F.-F.: Expansions of the exponential and the logarithm of expansions and applications. ResearchGate Research (2015). Available online at doi: 10.13140/RG.2.1.3309.2000
  11. 11.
    Temme, N.M.: Special Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, New York (1996). Available online at doi: 10.1002/9781118032572
  12. 12.
    Whittaker, E.T.; Watson, G.N.: A Course of Modern Analysis, reprint of the 4th edn (1927). Cambridge Mathematical Library. Cambridge University Press, Cambridge (1996). Available online at doi: 10.1017/CBO9780511608759

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsHenan Polytechnic UniversityJiaozuoChina
  2. 2.College of MathematicsInner Mongolia University for NationalitiesTongliaoChina
  3. 3.Department of Mathematics, College of ScienceTianjin Polytechnic UniversityTianjinChina

Personalised recommendations