  Recently, an analogue over $\mathbb{F}_q[T]$ of Landau's theorem on sums of
two squares was considered by Bary-Soroker, Smilansky and Wolf. They counted
the number of monic polynomials in $\mathbb{F}_q[T]$ of degree $n$ of the form
$A^2+TB^2$, which we denote by $B(n,q)$. They studied $B(n,q)$ in two limits:
fixed $n$ and large $q$; and fixed $q$ and large $n$. We generalize their
result to the most general limit $q^n \to \infty$. More precisely, we prove
\begin{equation*} B(n,q) \sim K_q \cdot \binom{n-\frac{1}{2}}{n} \cdot q^n ,
\qquad q^n \to \infty, \end{equation*} for an explicit constant $K_q =
1+O\left(1/q\right)$. Our methods are different and are based on giving
explicit bounds on the coefficients of generating functions. These methods also
apply to other problems, related to polynomials with prime factors of even
degree.
