# A note on Levi-Civita functional equation

## TL;DR

This paper investigates solutions to a generalized Levi-Civita functional equation on monoids, expanding understanding of its structure with multiple multiplicative function components.

## Contribution

It characterizes solutions to a complex functional equation involving sums of products of functions on monoids, generalizing classical results.

## Key findings

- Derived explicit solutions for the functional equation.
- Extended the class of known solutions to include multiple multiplicative functions.
- Provided conditions under which solutions exist.

## Abstract

In this paper we find the solutions of the functional equation $$f(xy) = g(x)h(y) + \sum_{j=1}^n g_j(x)h_j(y), \;x,y \in M,$$ where $M$ is a monoid, $n\geq 2$, and $g_j$ (for $j=1,...,n$) are linear combinations of at least $2$ distinct nonzero multiplicative functions.