[proofplan] The sheaf of holomorphic $E$-valued $p$-forms is resolved by the Dolbeault complex of smooth $E$-valued forms. The $\bar\partial$-Poincaré lemma gives local exactness, and smooth form sheaves are fine, hence acyclic. Taking global sections of this fine resolution computes sheaf cohomology and produces the Dolbeault cohomology groups. [/proofplan] [step:Write the Dolbeault resolution] Let $\mathcal{A}^{p,q}(E)$ be the sheaf of smooth $E$-valued $(p,q)$-forms. The holomorphic structure on $E$ defines an operator \begin{align*} \bar\partial_E:\mathcal{A}^{p,q}(E)\to\mathcal{A}^{p,q+1}(E). \end{align*} The kernel of $\bar\partial_E:\mathcal{A}^{p,0}(E)\to\mathcal{A}^{p,1}(E)$ is precisely $\mathcal{O}(E)\otimes_{\mathcal{O}_X}\Omega_X^p$. Thus we have a complex beginning with the desired holomorphic sheaf. [/step] [step:Use local exactness] On a holomorphic trivialising neighbourhood for $E$, the complex becomes a finite direct sum of ordinary Dolbeault complexes for scalar forms. The $\bar\partial$-Poincaré lemma is exact in positive degree on small polydiscs. Therefore \begin{align*} 0\to\mathcal{O}(E)\otimes\Omega_X^p\to\mathcal{A}^{p,0}(E)\xrightarrow{\bar\partial_E}\mathcal{A}^{p,1}(E)\xrightarrow{\bar\partial_E}\cdots \end{align*} is an exact sequence of sheaves. [/step] [step:Use fineness to compute cohomology] The sheaves $\mathcal{A}^{p,q}(E)$ are sheaves of smooth sections of vector bundles, so partitions of unity make them fine. Fine sheaves are acyclic. Hence the displayed Dolbeault complex is an acyclic resolution of $\mathcal{O}(E)\otimes\Omega_X^p$. [/step] [step:Identify global cohomology] Taking global sections of this acyclic resolution gives the complex \begin{align*} \Gamma(X,\mathcal{A}^{p,0}(E))\to\Gamma(X,\mathcal{A}^{p,1}(E))\to\cdots, \end{align*} whose cohomology is by definition $H^{p,q}_{\bar\partial}(X,E)$. The general theorem on cohomology computed by acyclic resolutions identifies this cohomology with $H^q(X,\mathcal{O}(E)\otimes\Omega_X^p)$. Taking $p=0$ gives the stated special case. [/step]