Correlation of multiplicative functions over function fields

In this article we study the asymptotic behaviour of the correlation functions over polynomial ring ${\mathbb{F}}_q[x]$. Let ${\mathcal{M}}_{n,q}$ and ${\mathcal{P}}_{n,q}$ be the set of all monic polynomials and monic irreducible polynomials of degree $n$ over ${\mathbb{F}}_q$ respectively. For multiplicative functions ${\psi }_1$ and ${\psi }_2$ on ${\mathbb{F}}_q[x]$, we obtain asymptotic formula for the following correlation functions for a fixed $q$ and $n\to \infty$ \begin{align*} &S_{2}(n, q):=\displaystyle\sum_{f\in $\mathcal{M}${n, q}}\psi_1(f+h_1) \psi_2(f+h_2), \ &R_2(n, q):=\displaystyle\sum{P\in $\mathcal{P}${n, q}}\psi_1(P+h_1)\psi_2(P+h_2), \end{align*} where $h_1,h_2$ are fixed polynomials of degree $<n$ over ${\mathbb{F}}_q$. As a consequence, for real valued additive functions $\widetilde{{\psi }_1}$ and $\widetilde{{\psi }_2}$ on ${\mathbb{F}}_q[x]$ we show that for a fixed $q$ and $n\to \mathrm{\infty }$, the following distribution functions \begin{align*} &\frac{1}{|$\mathcal{M}${n, q}|}\Big|{f\in $\mathcal{M}${n, q} : \tilde{\psi_1}(f+h_1)+\tilde{\psi_2}(f+h_2)\leq x}\Big|,\ & \frac{1}{|$\mathcal{P}${n, q}|}\Big|{P\in $\mathcal{P}$_{n, q} : \tilde{\psi_1}(P+h_1)+\tilde{\psi_2}(P+h_2)\leq x}\Big| \end{align*} converges weakly towards a limit distribution.