Hypertranscendental Formal Power Series Over Fields of Positive Characteristic

Let kk be a field of characteristic p>0p>0, k[[x]]k[[x]], the ring of formal power series over kk, k((x))k((x)), the quotient field of k[[x]]k[[x]], and k(x)k (x) the field of rational functions over kk. we shall give some characterizations of an algebraic function f∈k((x))f∈k ((x)) over kk. let ll be a field of characteristic zero. the power series f∈l[[x]]f∈l[[x]] is called differentially algebraic, if it satisfies a differential equation of the form p(x,y,y′,...)=0p(x,y,y ′,...)=0, where pp is a non-trivial polynomial. this notion is defined over fields of characteristic zero and is not so significant over fields of characteristic p>0p>0, since f(p)=0f(p)=0. we shall define an analogue of the concept of a differentially algebraic power series over kk and we shall find some more related results.