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Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field$\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in$\mathbb{A}$. In this paper, we show that a large class of arithmetic functionsin multi-variables over $\mathbb{A}_{+}$ can be expanded through the polynomialRamanujan sums and the unitary polynomial Ramanujan sums. These are analoguesof classical results over $\mathbb{N}$ by Winter, Delange and T\'oth.

Ramanujan expansions of arithmetic functions of several variables over $\mathbb{F}_{q}[T]$