  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 $\Fq[[T]]$ in one
variable $T$ over a finite field $\Fq$ of $q$ elements. Accordingly, we show
that two sequences of Cartier operators are an orthonormal basis of the space
of continuous $\Fq$-linear functions on $\Fq[[T]].$ According to the digit
principle, every continuous function on $\Fq[[T]]$ 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 $\Zp.$
