hypertranscendental formal power series over fields of positive characteristic

let $k$ be a field of characteristic$p>0$, $k[[x]]$, the ring of formal power series over $ k$,$k((x))$, the quotient field of $ k[[x]]$, and $ k(x)$ the fieldof rational functions over $k$. we shall give somecharacterizations of an algebraic function $fin k((x))$ over $k$.let $l$ be a field of characteristic zero. the power series $finl[[x]]$ is called differentially algebraic, if it satisfies adifferential equation of the form $p(x, y, y',...)=0$, where $p$is a non-trivial polynomial. this notion is defined over fields ofcharacteristic zero and is not so significant over fields ofcharacteristic $p>0$, since $f^{(p)}=0$. we shall define ananalogue of the concept of a differentially algebraic power seriesover $k$ and we shall find some more related results.