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From an analytical perspective, we introduce a sequence of Cartier operatorsthat act on the field of formal Laurent series in one variable withcoefficients in a field of positive characteristic $p$. In this work, wediscover the binomial inversion formula between Hasse derivatives and Cartieroperators, implying that Cartier operators can play a prominent role in variousobjects of study in function field arithmetic, as suitable substitutes forhigher derivatives. For an applicable object, the Wronskian criteria associatedwith Cartier operators are introduced. These results stem from a careful studyof two types of Cartier operators on the power series ring $\Fq[[T]]$ in onevariable $T$ over a finite field $\Fq$ of $q$ elements. Accordingly, we showthat two sequences of Cartier operators are an orthonormal basis of the spaceof continuous $\Fq$-linear functions on $\Fq[[T]].$ According to the digitprinciple, every continuous function on $\Fq[[T]]$ is uniquely written in termsof a $q$-adic extension of Cartier operators, with a closed-form of expansioncoefficients for each of the two cases. Moreover, the $p$-adic analogues ofCartier operators are discussed as orthonormal bases for the space ofcontinuous functions on $\Zp.$

Cartier operators on fields of positive characteristic p