TITLE: Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle
QUESTION [7 upvotes]: I am trying to understand the jet bundles but currently I am stuck on the following questions: 
Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) manifold $X$. I know that the bundle $J_k(E)$ of k-jets of $E$ has the structure of a vector bundle over $X$.
I would like to know however:
1.) What is the rank of the vector bundle $J_k(E)$ ? 
2.) Is $J_k(E)$ holomorphic in the case when $(E, \pi, X)$ is holomorphic? 
3.) When $\pi: E\rightarrow X$ is a fiber bundle with structure group $G$, can we view $J_1(E)$ as the associated principal $G$-bundle $P$ associated to $E$ or not?
I have seen an interpretation of $J_1(E)$ as some sort of an "extended frame bundle" of E in the sense that its fiber consists of the set of all pairs comprising a basis of $T_pX$   $(T^{1, 0}_pX)$ and a basis of $E_p$, $p\in X$.
P.S.: I am new here and I really hope that I don't annoy the experienced audience in this forum with trivialities. I would appreciate any help or suggestions or simply good references. Thank you in advance for your competent help.

REPLY [2 votes]: Please let me add something about the question of holomorphicity:
If $E$ is a holomorphic vector bundle, then there is a bundle of jets of holomorphic sections, which is holomorphic, as explained by @hm2020. The bundle of 1-jets of smooth sections of $E$, on the other side, does not have a holomorphic structure.
Example: let $E$ be a trivial bundle over $X$, so that the exact sequence described by @abx splits. Then the jet bundle of smooth sections of $E$ is
$$(T^*_{\mathbb R}X\otimes_{\mathbb R} E) \oplus E = (\Omega^{1,0}(X)\otimes_{\mathbb C} E) \oplus (\Omega^{0,1}(X)\otimes_{\mathbb C} E) \oplus E~.$$
While $(\Omega^{1,0}(X)\otimes E)\oplus E$ has a holomorphic structure and identifies with the bundle of jets of holomorphic sections, the other summand $\Omega^{0,1}(X)$ does not have a preferred holomorphic structure.