= Solution
For a <holomorphic vector bundle> $E$, the <sheaf of vector-bundle-valued differential forms> is
$$
\mathcal A^{p,q}(E)(U)=\Gamma\left(U,
\bigwedge^p(T^{1,0}X)^*\otimes\bigwedge^q(T^{0,1}X)^*\otimes E\right),
$$
where sections are smooth and restrictions are the usual ones. In a <holomorphic local frame> of $E$, the <Dolbeault operator> $\bar\partial_E$ acts coefficientwise and satisfies $\bar\partial_E^2=0$. Its <Dolbeault cohomology with values in a holomorphic vector bundle> is the cohomology of the global section complex $\mathcal A^{p,\bullet}(X,E)$.
The bundle-valued <Dolbeault theorem> identifies this with <sheaf cohomology>:
$$
\boxed{H^q(X,\Omega_X^p\otimes\mathcal O(E))\cong H^{p,q}_{\bar\partial}(X,E).}
$$
Here $\Omega_X^p$ is the <sheaf of holomorphic differential forms>, and $\mathcal O(E)$ is the <sheaf of holomorphic sections of a vector bundle>. Equivalently, the complex
$$
0\longrightarrow\Omega_X^p\otimes\mathcal O(E)
\longrightarrow\mathcal A^{p,0}(E)\xrightarrow{\bar\partial_E}\mathcal A^{p,1}(E)
\xrightarrow{\bar\partial_E}\cdots
$$
is a <fine sheaf> resolution, by the local <Dolbeault-Poincaré lemma> and smooth <partitions of unity>. In particular $p=0$ computes $H^q(X,\mathcal O(E))$; for $p>0$ the holomorphic differential-form factor must be retained.
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