# Some notes on the algebraic structure of linear recurrent sequences

## TL;DR

This paper explores the algebraic structures of linear recurrent sequences under various operations, proving they form R-algebras and providing explicit characteristic polynomials, with an investigation into their isomorphisms.

## Contribution

It establishes that sets of linear recurrent sequences form R-algebras under certain operations and explicitly characterizes their algebraic properties.

## Key findings

- Linear recurrent sequences form R-algebras with sum and product operations.
- Explicit characteristic polynomials for Hurwitz and Newton products are provided.
- Investigation into isomorphisms between different R-algebras is conducted.

## Abstract

Several operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products are R-algebras, given any commutative ring $R$ with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these $R-$algebras are isomorphic, considering also the R-algebras obtained using the Hadamard product and the convolution product.