  For any positive integer $n$, we introduce a quasi-homogeneous vector field
$\textsf{D}$ of degree $2$ on a moduli space $\textsf{T}$ of enhanced
Calabi-Yau $n$-folds arising from the Dwork family. By Calabi-Yau quasi-modular
forms for Dwork family we mean the elements of the graded $\mathbb{C}$-algebra
$\widetilde{\mathcal{M}}$ generated by the components of a particular solution
of $\textsf{D}$, which are provided with natural weight. Using $\textsf{D}$ we
introduce the derivation $\mathcal{D}$ and the Ramanujan-Serre type derivation
$\partial$ on $\widetilde{\mathcal{M}}$. We show that they are degree $2$
differential operators and there exists a proper subspace $\mathcal{M}\subset
\widetilde{\mathcal{M}}$, called the space of Calabi-Yau modular forms, which
is closed under $\partial$. Using the derivation $\mathcal{D}$, we define the
Rankin-Cohen brackets for Calabi-Yau quasi-modular forms and prove that the
subspace generated by the positive weight elements of $\mathcal{M}$ is closed
under the Rankin-Cohen brackets.
