This question comes from a paper of Vojta "A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing":
https://www.jstor.org/stable/2152710?seq=23#metadata_info_tab_contents
Here is the situation: Let $C$ be a curve defined over a number field $k$, $B$ be the arithmetic scheme corresponding to $k$ ($B = \operatorname{Spec} R$ where $\operatorname{Frac}(R)=k$). Let $\pi: X \to B$ be an arithmetic surface assumed to be regular (viewed as a regular model of $C$).
On page 785, he defined the partial derivative of a section as follows:
Let $\gamma$ be a section of a metrized line sheaf $\mathcal{L}$ on $X$, and let $P\in C(k)$, corresponding to a section $s: B \to X$. Then for integers $i\geq 0$, we shall now define the $i$-th "derivative" $D_{i}\gamma(P)$ as a section: $$D_{i}\gamma \in \Gamma(B,s^{*}(\mathcal{L}\otimes\omega_{X/B}^{\otimes i})),$$ provided that $D_{j}(\gamma)=0$ for all $j=0,\ldots,i-1$. Fix a local section $\gamma_{0}$ of $\mathcal{L}$ which has neither a zero nor a pole at $P$; then $$\gamma=\gamma_{0}\sum_{i \geq 0}a_{i}z^{i}$$ and $\gamma_{0}a_{i}z^{i}$ lies in the subsheaf $$\mathcal{L}\otimes \mathcal{O}(-iP)\subset \mathcal{L}.$$ Its restriction to the image of $s$ is well-defined, independent of the choices of $z$ and $\gamma_{0}$, by the assumption on $i$. Using different local parameters $z$, it follows that this section is a well-defined element of the sheaf $s^{*}(\mathcal{L}\otimes \mathcal{O}(-iP))\cong s^{*}(\mathcal{L}\otimes \omega_{X/B}^{\otimes i})$ at every closed point of $B$.
It looks like the derivative section is chosen to be $\gamma_{0}a_{i}z^{i}$? I am not sure what he means here.
Another thing confusing me is the condition $D_{j}\gamma = 0$ for $j=0,\ldots,i-1$. How can one say something about $D_{j}\gamma$ before defining it?
I am just having trouble understanding what this paragraph means. Can anyone give some explanations/comments?
Thank you!