A Menon—Sury-type identity for arithmetic functions on $\mathbb{F}_q[T]$

Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over the finite field $\mathbb{F}_{q}$. In this paper, we prove a Menon—Sury's identity with several multiplicative and additive characters for any arithmetic function $f$ on $\mathbb{A}$. Our result implies several interesting identities which may be viewed as the analogues of known ones in $\mathbb{Z}$.