Cartier operators on fields of positive characteristic p

From an analytical perspective, we introduce a sequence of Cartier operators that act on the field of formal Laurent series in one variable with coefficients in a field of positive characteristic $p$. In this work, we discover the binomial inversion formula between Hasse derivatives and Cartier operators, implying that Cartier operators can play a prominent role in various objects of study in function field arithmetic, as suitable substitutes for higher derivatives. For an applicable object, the Wronskian criteria associated with Cartier operators are introduced. These results stem from a careful study of two types of Cartier operators on the power series ring $\text{Fq}[[T]]$ in one variable $T$ over a finite field $\text{Fq}$ of $q$ elements. Accordingly, we show that two sequences of Cartier operators are an orthonormal basis of the space of continuous $\text{Fq}$-linear functions on $\text{Fq}[[T]]\mathrm{.}$ According to the digit principle, every continuous function on $\text{Fq}[[T]]$ is uniquely written in terms of a $q$-adic extension of Cartier operators, with a closed-form of expansion coefficients for each of the two cases. Moreover, the $p$-adic analogues of Cartier operators are discussed as orthonormal bases for the space of continuous functions on $\text{Zp}\mathrm{.}$