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

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.