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Results in Mathematics

, 73:161 | Cite as

Harmonic Number Expansions of the Ramanujan Type

  • Weiping Wang
Article
  • 8 Downloads

Abstract

In this paper, we establish three (general) asymptotic expansions of the Ramanujan type for the harmonic numbers, and give the corresponding recurrences of the coefficient sequence or parameter sequences in these expansions. We also present two explicit expressions for the coefficient sequence of the first expansion by the methods of generating functions and summation transformations. It can be found that the first expansion includes the classical Ramanujan formula, the DeTemple–Wang formula and the Chen–Mortici–Villarino formula as special cases, and the third one includes the refinement of Lodge’s approximation as a special case. Moreover, the second and third expansions are lacunary and contain only even power terms or odd power terms. By these expansions, we give unified approaches to dealing with asymptotic expansions of the Ramanujan type for the harmonic numbers.

Keywords

Asymptotic expansions combinatorial identities harmonic numbers 

Mathematics Subject Classification

41A60 05A19 11B83 

Notes

Acknowledgements

The author is supported by the National Natural Science Foundation of China (under Grant 11671360), and the Zhejiang Provincial Natural Science Foundation of China (under Grant LQ17A010010).

References

  1. 1.
    Berndt, B.C.: Ramanujan’s Notebooks. Part V. Springer, New York (1998)CrossRefGoogle Scholar
  2. 2.
    Burić, T., Elezović, N.: Approximants of the Euler–Mascheroni constant and harmonic numbers. Appl. Math. Comput. 222, 604–611 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cao, X.: Multiple-correction and continued fraction approximation. J. Math. Anal. Appl. 424(2), 1425–1446 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cao, X., Xu, H., You, X.: Multiple-correction and faster approximation. J. Number Theory 149, 327–350 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, C.-P.: On the coefficients of asymptotic expansion for the harmonic number by Ramanujan. Ramanujan J. 40(2), 279–290 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, C.-P.: Stirling expansions into negative powers of a triangular number. Ramanujan J. 39(1), 107–116 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, C.-P., Cheng, J.-X.: Ramanujan’s asymptotic expansion for the harmonic numbers. Ramanujan J. 38(1), 123–128 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, C.-P., Mortici, C.: New sequence converging towards the Euler–Mascheroni constant. Comput. Math. Appl. 64(4), 391–398 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    DeTemple, D.W., Wang, S.-H.: Half integer approximations for the partial sums of the harmonic series. J. Math. Anal. Appl. 160(1), 149–156 (1991)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Feng, L., Wang, W.: Riordan array approach to the coefficients of Ramanujan’s harmonic number expansion. Results Math. 71(3–4), 1413–1419 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hirschhorn, M.D.: Ramanujan’s enigmatic formula for the harmonic series. Ramanujan J. 27(3), 343–347 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Issaka, A.: An asymptotic series related to Ramanujan’s expansion for the \(n\)th harmonic number. Ramanujan J. 39(2), 303–313 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lu, D.: A new quicker sequence convergent to Euler’s constant. J. Number Theory 136, 320–329 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lu, D.: Some quicker classes of sequences convergent to Euler’s constant. Appl. Math. Comput. 232, 172–177 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lu, D., Song, L., Yu, Y.: Some new continued fraction approximation of Euler’s constant. J. Number Theory 147, 69–80 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lodge, A.: An approximate expression for the value of \(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{r}\). Messenger Math. 30, 103–107 (1904)Google Scholar
  17. 17.
    Mortici, C.: On new sequences converging towards the Euler–Mascheroni constant. Comput. Math. Appl. 59(8), 2610–2614 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mortici, C.: On the Stirling expansion into negative powers of a triangular number. Math. Commun. 15(2), 359–364 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mortici, C., Chen, C.-P.: On the harmonic number expansion by Ramanujan. J. Inequal. Appl. 2013, 222 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mortici, C., Villarino, M.B.: On the Ramanujan–Lodge harmonic number expansion. Appl. Math. Comput. 251, 423–430 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Nemes, G.: Asymptotic expansion for \(\log n!\) in terms of the reciprocal of a triangular number. Acta Math. Hung. 129(3), 254–262 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nemes, G.: More accurate approximations for the gamma function. Thai J. Math. 9(1), 21–28 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Shapiro, L.W., Getu, S., Woan, W.J., Woodson, L.C.: The Riordan group. Discrete Appl. Math. 34(1–3), 229–239 (1991)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sprugnoli, R.: Riordan arrays and combinatorial sums. Discrete Math. 132(1–3), 267–290 (1994)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Villarino, M.B.: Ramanujan’s harmonic number expansion into negative powers of a triangular number. JIPAM. J. Inequal. Pure Appl. Math. 9(3), 89 (2008)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Wang, W.: Unified approaches to the approximations of the gamma function. J. Number Theory 163, 570–595 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Xu, H., You, X.: Continued fraction inequalities for the Euler–Mascheroni constant. J. Inequal. Appl. 2014, 343 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Yang, S.: On an open problem of Chen and Mortici concerning the Euler–Mascheroni constant. J. Math. Anal. Appl. 396(2), 689–693 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    You, X., Chen, D.-R.: Some sharp continued fraction inequalities for the Euler–Mascheroni constant. J. Inequal. Appl. 2015, 308 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of ScienceZhejiang Sci-Tech UniversityHangzhouChina

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