A generalization of Tóth identity in $\mathbb{F}_q[T]$ involving a Dirichlet Character

Abstract: Let $\mathbb{A}=\mathbb{F}_q[T]$ be the polynomial ring over the finite field $\mathbb{F}_q$. In this article, we prove a generalization of T\'oth identity on $\mathbb{A}$ involving arithmetical functions, multiplicative and additive characters.