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Research in Number Theory

, 4:28 | Cite as

An isomorphism between the convolution product and the componentwise sum connected to the D’Arcais numbers and the Ramanujan tau function

  • Stefano Barbero
  • Umberto Cerruti
  • Nadir Murru
Research

Abstract

Given a commutative ring R with identity, let \(H_R\) be the set of sequences of elements in R. We investigate a novel isomorphism between \((H_R, +)\) and \((\tilde{H}_R,*)\), where \(+\) is the componentwise sum, \(*\) is the convolution product (or Cauchy product) and \(\tilde{H}_R\) the set of sequences starting with \(1_R\). We also define a recursive transform over \(H_R\) that, together to the isomorphism, allows to highlight new relations among some well studied integer sequences. Moreover, these connections allow to introduce a family of polynomials connected to the D’Arcais numbers and the Ramanujan tau function. In this way, we also deduce relations involving the Bell polynomials, the divisor function and the Ramanujan tau function. Finally, we highlight a connection between Cauchy and Dirichlet products.

Keywords

Bell polynomials Convolution product D’Arcais numbers Ramanujan tau function 

Mathematics Subject Classification

11A25 11B75 11T06 13F25 

Notes

Author's contributions

The authors would like to express their gratitude to the referee for a very careful reading of the article, and valuable comments, which have improved the article.

References

  1. 1.
    Barbero, S., Cerruti, U., Murru, N.: Some combinatorial properties of the Hurwitz series ring. Ricer. Mat. (2017).  https://doi.org/10.1007/s11587-017-0336-x
  2. 2.
    Barbero, S., Cerruti, U., Murru, N.: On the operations over sequences in rings and binomial type sequences. Ricer. Mat. (2018).  https://doi.org/10.1007/s11587-018-0389-5
  3. 3.
    Baruah, N.D., Sarmah, B.K.: Identities and congruences for the general partition and Ramanujan’s tau functions. Indian J. Pure Appl. Math. 44, 643–671 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benhissi, A.: Ideal structure of Hurwitz series rings. Contrib. Algebra Geom. 48, 251–256 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Benhissi, A., Koja, F.: Basic properties of Hurwitz series rings. Ricer. Mat. 61, 255–273 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Charles, D.X.: Computing the Ramanujan tau function. The Ramanujan J. 11(2), 221–224 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Comtet, L.: Advanced Combinatorics. Reidel, Boston (1974)CrossRefGoogle Scholar
  8. 8.
    Ghanem, M.: Some properties of Hurwitz series ring. Int. Math. Forum 6(40), 1973–1981 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Keigher, W.F.: On the ring of Hurwitz series. Commun. Algebra 25, 1845–1859 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Keigher, W.F., Pritchard, F.L.: Hurwitz series as formal functions. J. Pure Appl. Algebra 146, 291–304 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lygeros, N., Rozier, O.: Odd prime values of the Ramanujan tau function. The Ramanujan J. 32(2), 269–280 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mordell, J.L.: On Mr. Ramanujan’s empirical expansions of modular functions. Proc. Camb. Philos. Soc. 19, 117–124 (1917)zbMATHGoogle Scholar
  13. 13.
    Ramanujan, S.: On certain arithmetical functions. Trans. Camb. Philos. Soc. 22(9), 159–184 (1916)Google Scholar
  14. 14.
    Singh, J.: On an arithmetic convolution. J. Integer Seq. 17(2), 3 (2014)MathSciNetGoogle Scholar
  15. 15.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/~njas/sequences (2010)
  16. 16.
    Tóth, L., Haukkanen, P.: On the binomial convolution of arithmetical functions. J. Comb. Number Theory 1(1), 31–48 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of Mathematics G. PeanoUniversity of TurinTorinoItaly

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