  This paper deals with function field analogues of the famous theorem of
Landau which gives the asymptotic density of sums of two squares in
$\mathbb{Z}$.
  We define the analogue of a sum of two squares in $\mathbb{F}_q[T]$ and
estimate the number $B_q(n)$ of such polynomials of degree $n$ in two cases.
The first case is when $q$ is large and $n$ fixed and the second case is when
$n$ is large and $q$ is fixed. Although the methods used and main terms
computed in each of the two cases differ, the two iterated limits of (a
normalization of) $B_q(n)$ turn out to be exactly the same.
