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. 



Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle
(1) Locally, jets of sections are just collections of $r=\operatorname{rank}E$ jets of functions, hence, the rank of $J\_k(E)$ equals $r$ times the number of multiindices $I=(i\_1,\ldots,i\_n)$ with $|I|\le k$. 


(2) It is certainly holomorphic.


(3) It seems to me that $J\_1(E)=T^\*X\otimes E$.


