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 MurruEmail author


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.


Bell polynomials Convolution product D’Arcais numbers Ramanujan tau function 

Mathematics Subject Classification

11A25 11B75 11T06 13F25 


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.


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Copyright information

© SpringerNature 2018

Authors and Affiliations

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

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