Published April 30, 2024 | Version CC BY-NC-ND 4.0
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On Some Relations Involving the Ramanujan's Tau Function

Creators

  • 1. Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai (Tamil Nadu), India.

Contributors

Contact person:

  • 1. Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai (Tamil Nadu), India.
  • 2. ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 0778, CDMX, México.

Description

Abstract: It is known a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function𝝈(𝒏), whose solution gives a closed formula for 𝝉(𝒏) in terms ofcomplete Bell polynomials, and a determinantal expression for 𝝈(𝒎) where participate the values 𝝉(𝒌).

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Additional details

Identifiers

DOI
10.54105/ijam.A1168.04010424
EISSN
2582-8932

Dates

Accepted
2024-04-15
Manuscript received on 18 March 2024 | Revised Manuscript received on 21 March 2024 | Manuscript Accepted on 15 April 2024 | Manuscript published on 30 April 2024

References

  • S. Ramanujan, On certain arithmetical functions, Trans. Camb. Phil. Soc. 22, No. 9 (1916) 159-184.
  • R. Roy, Elliptic and modular functions from Gauss to Dedekind to Hecke, Cambridge University Press (2017). https://doi.org/10.1017/9781316671504
  • http://oeis.org
  • L. J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proc. Cambridge Phil. Soc. 19 (1917) 117-124.
  • P. Deligne, La conjecture of Weil, Pub. Mathématiques de l'IHÉS 43, No. 3 (1974) 273-307. https://doi.org/10.1007/BF02684373
  • T. M. Apostol, Modular functions and Dirichlet series in number theory, Springer-Verlag, New York (1997).
  • Wen-Ching Winnie Li, The Ramanujan conjecture and its applications, Phil. Trans. R. Soc. A378(2019) 20180441. https://doi.org/10.1098/rsta.2018.0441
  • R. Sivaraman, J. López-Bonilla, D. Morales-Cruz, S. Vidal-Beltrán, On specially multiplicative functions, European J. of Appl. Sci. Eng. & Tech. 2, No. 1 (2024) 21-25. https://doi.org/10.59324/ejaset.2024.2(1).03
  • K. S. Kölbig, The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind, J. Comput. Appl. Math. 51 (1994) 113-116. https://doi.org/10.1016/0377-0427(94)00010-7
  • W. P. Johnson, The curious history of Faä di Bruno's formula, The Math. Assoc. of America 109 (2002) 217-234. https://doi.org/10.1080/00029890.2002.11919857
  • D. F. Connon, Various applications of the (exponential) complete Bell polynomials, http://arxiv.org/ftp/arkiv/papers/1001/1001.2835.pdf , 16 Jan 2010.
  • J. López-Bonilla, S. Vidal-Beltrán, A. Zúñiga-Segundo, Some applications of complete Bell polynomials, World Eng. & Appl. Sci. J. 9, No. 3 (2018) 89-92.
  • J. López-Bonilla, R. López-Vázquez, S. Vidal-Beltrán, Bell polynomials, Prespacetime J. 9, No. 5 (2018) 451-453.
  • J. López-Bonilla, S. Vidal-Beltrán, A. Zúñiga-Segundo, Characteristic equation of a matrix via Bell polynomials, Asia Mathematika 2, No. 2 (2018) 49-51.
  • R. Sivaraman, J. D. Bulnes, J. López-Bonilla, Complete Bell polynomials and recurrence relations for arithmetic functions, European J. of Theor. Appl. Sci. 1, No. 3 (2023) 167-170. https://doi.org/10.59324/ejtas.2023.1(3).18
  • J. C. Mason, D. C. Handscomb, Chebyshev polynomials, Chapman & Hall / CRC, London (2002). https://doi.org/10.1201/9781420036114
  • J. Ewell, New representations of Ramanujan's tau function, Proc. Amer. Math. Soc. 128 (1999) 723-726. https://doi.org/10.1090/S0002-9939-99-05289-2
  • Sivaraman, Dr. R., Núñez-Yépez, H. N., & López-Bonilla, Prof. J. (2023). Ramanujan's Tau-Function in Terms of Bell Polynomials. In Indian Journal of Advanced Mathematics (Vol. 3, Issue 2, pp. 1–3). https://doi.org/10.54105/ijam.b1157.103223
  • Bashir, S. (2023). Pedagogy of Mathematics. In International Journal of Basic Sciences and Applied Computing (Vol. 10, Issue 2, pp. 1– 8). https://doi.org/10.35940/ijbsac.b1159.1010223
  • Rao, K. N. B., Srinivas, Dr. G., & D., Dr. P. R. P. V. G. (2019). A Heuristic Ranking of Different Characteristic Mining Based Mathematical Formulae Retrieval Models. In International Journal of Engineering and Advanced Technology (Vol. 9, Issue 1, pp. 893– 901). https://doi.org/10.35940/ijeat.a9412.109119
  • Lal, M., Jha, Y. D., & Zuberi, H. A. (2020). A Two Phase Mathematical Model of Fluid Flow through Bell Shaped Stenotic Artery. In International Journal of Innovative Technology and Exploring Engineering (Vol. 9, Issue 5, pp. 35–38). https://doi.org/10.35940/ijitee.e1966.039520
  • Tahiliani, Dr. S. (2021). More on Diophantine Equations. In International Journal of Management and Humanities (Vol. 5, Issue 6, pp. 26–27). https://doi.org/10.35940/ijmh.l1081.025621