  Bateman and Duquette have initiated the study of Salem elements in positive
characteristic. This work extends their results to 2-Salem elements whose
minimal polynomials are of the type $Y^n + \lambda_{n-1}Y^{n-1} +
\ldots+\lambda_1Y + \lambda_0 \in \mathbb{F}_q[X][Y ]$ where $n \geq2,
\lambda_0\neq 0$ and $\deg \lambda_{n-1} < \deg \lambda_{n-2} = \max_{i\neq
n-2}\deg(\lambda_i)$. This work provides an analogue of their results for
2-Salem elements whose minimal polynomials meet certain requirements.
