Let $f=\sum_{i=0}^\infty f_i T^i\in \mathbb{F}[[T]]$ be a formal power series over a field $\mathbb{F}$ of characteristic $p$. For every $k\geq 0$, we define the $k$-th Hasse derivative on $\mathbb{F}[[T]]$: $$\partial_T^{[k]}\colon f\longmapsto \sum_{i=0}^\infty f_i\binom{i}{k}T^{i-k}.$$ Then an easy observation is that we can extract the coefficients with the Hasse derivation: $\left(\partial_T^{[k]}f\right)(0)=f_k$. And this gives a Taylor expansion formula for formal power series over field of positive characteristic. Evenmore, we can generalize this to the multivariable case.
Now my question is that, can we extract the coefficients of formal Laurent series $g=\sum_{i>\!>-\infty}g_i T^i\in\mathbb{F}((T))$ by using a similar operator? If we replace $\mathbb{F}$ with $\mathbb{C}$, then the Cauchy integrate formula may give what we want. However there is no such thing in characteristic $p$.