View More View Less
  • 1 Loránd Eötvös University, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary
Restricted access

Abstract

Ramanujan suggested an expansion for the nth partial sum of the harmonic series which employs the reciprocal of the nth triangular number. This has been proved in 2006 by Villarino, who speculated that there might also exist a similar expansion for the logarithm of the factorial. This study shows that such an asymptotic expansion indeed exists and provides formulas for its generic coefficient and for the bounds on its errors.

  • [1] Abramowitz, M., Stegun, I. A. (eds.) 1965 Handbook of Mathematical Functions Dover New York.

  • [2] Allasia, G., Giordano, C., Pečarić, J. 2002 Inequalities for the Gamma function relating to asymptotic expansions Math. Inequal. Appl. 5 543555.

    • Search Google Scholar
    • Export Citation
  • [3] Alzer, H. 2003 On Ramanujan’s double-inequality for the gamma function Bull. London Math. Soc. 35 243249 .

  • [4] Berndt, B. 1998 Ramanujan’s Notebooks 5 Springer New York 531532.

  • [5] Copson, E. T. 1965 Asymptotic Expansions Cambridge University Press Cambridge .

  • [6] Karatsuba, E. A. 2001 On the asymptotic representation of the Euler gamma function by Ramanujan J. Comput. Appl. Math. 135 225240 .

  • [7] Ragahavan, S., Rangachari, S. S. (eds.) 1988 S. Ramanujan: The Lost Notebook and Other Unpublished Papers Narosa New Delhi.

  • [8] Stirling, J., Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium (London, 1730), 136137.

    • Search Google Scholar
    • Export Citation
  • [9] Villarino, M. B. 2008 Ramanujan’s harmonic number expansion into negative powers of a triangular number J. Inequal. Pure and Appl. Math. 9 112.

    • Search Google Scholar
    • Export Citation

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Sep 2020 1 0 0
Oct 2020 0 0 0
Nov 2020 1 7 2
Dec 2020 3 0 0
Jan 2021 1 0 0
Feb 2021 2 0 0
Mar 2021 0 0 0