  Let $\mathbb{F}_{q}$ be the finite field with $q$ elements. This paper mainly
researches the polynomial representation of double cyclic codes over
$\mathbb{F}_{q}+v\mathbb{F}_{q}+v^2\mathbb{F}_{q}$ with $v^3=v$. Firstly, we
give the generating polynomials of these double cyclic codes. Secondly, we show
the generating matrices of them. Meanwhile, we get quantitative information
related to them by the matrix forms. Finally, we investigate the relationship
between the generators of double cyclic codes and their duals.
