# Convolutions of Liouvillian Sequences

## TL;DR

This paper investigates the closure properties of Liouvillian and d'Alembertian sequences under convolution, revealing specific conditions under which these sequences are closed under this operation.

## Contribution

It establishes that Liouvillian sequences are closed under convolution with rationally Liouvillian sequences, and d'Alembertian sequences are closed under convolution with rationally d'Alembertian sequences.

## Key findings

- Liouvillian sequences are not closed under convolution in general.
- d'Alembertian sequences are not closed under convolution in general.
- Closure under convolution holds when sequences are combined with rationally related sequences.

## Abstract

While Liouvillian sequences are closed under many operations, simple examples show that they are not closed under convolution, and the same goes for d'Alembertian sequences. Nevertheless, we show that d'Alembertian sequences are closed under convolution with rationally d'Alembertian sequences, and that Liouvillian sequences are closed under convolution with rationally Liouvillian sequences.