  A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal
C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ is called cyclic if
the set of coordinates can be partitioned into two subsets, the set of
${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such that any
simultaneous cyclic shift of the coordinates of both subsets leaves invariant
the code. These codes can be identified as submodules of the
$\mathbb{Z}_4[x]$-module
$\mathbb{Z}_2[x]/(x^\alpha-1)\times\mathbb{Z}_4[x]/(x^\beta-1)$. Any
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic code ${\cal C}$ is of the form
$\langle (b(x)\mid{ 0}), (\ell(x) \mid f(x)h(x) +2f(x)) \rangle$ for some
$b(x), \ell(x)\in\mathbb{Z}_2[x]/(x^\alpha-1)$ and $f(x),h(x)\in
{\mathbb{Z}}_4[x]/(x^\beta-1)$. A new algorithm is presented to compute the
generator polynomials for ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic codes.
