A note on Levi-Civita functional equation

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, $n\ge 2$, and $g_j$ (for $j=1,...,n$) are linear combinations of at least $2$ distinct nonzero multiplicative functions.

14 pages