Algebra of formal power series, isomorphic to the algebra of formal Dirichlet series

E. Burlachenko

Published 2017-02-03Version 1

Ordinary algebra of formal power series in one variable is convenient to study by means of the algebra of Riordan matrices and the Riordan group. In this paper we consider algebra of formal power series without constant term, isomorphic to the algebra of formal Dirichlet series. To study it, we introduce matrices, similar to the Riordan matrices. As a result, some analogies between two algebras becomes visible. For example, the Bell polynomials (polynomials of partitions of number $n$ into $m$ parts) play a certain role in the ordinary algebra. Similar polynomials (polynomials of decompositions of number $n$ into $m$ factors) play a similar role in the considered algebra. Analog of the Lagrange series for the considered algebra is also exists. In connection with this analogy, we introduce matrix group, similar to the Riordan group and called the Riordan-Dirichlet group. As an example, we consider analog of the Abel's identities for this group.