The number of prime factors in $h$-free and $h$-full polynomials over function fields

We study the distribution of the number of prime divisors function $\Omega$ over polynomials of the function field $\mathbb{F}_q(T)$ when restricted to $h$-free polynomials and to $h$-full polynomials. We use an adaptation of the Selberg—Delange method to arithmetical semigroups due to Warlimont to compute the first and second moments in each case and show that a generalization of the Erdős—Kac Theorem is true.