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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 a suitable substitute 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 $\Zp.$
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참고문헌 (32)
참고문헌 신청
J. Allouche / 2003 / Automatic sequences, Theory, applications, generalizations / Cambridge University Press
B. Angles / 2015 / Universal Gauss-Thakur sums and L-series / Invent. Math. 200(2):653~669
V. Bosser / 2008 / Hyperdifferential properties of Drinfeld quasi-modular forms / Int. Math. Res. Not. IMRN 2008(11):56
A. Bostan / 2010 / Wronkians and linear independence / Amer. Math. Monthly 117(8):722~727
L. Carlitz / 1940 / A set of polynomials / Duke Math. J. 6:486~504
B. Angles / 2015 / Universal Gauss-Thakur sums and L-series / Invent. Math. 200(2):653~669
V. Bosser / 2008 / Hyperdifferential properties of Drinfeld quasi-modular forms / Int. Math. Res. Not. IMRN 2008(11):56
A. Bostan / 2010 / Wronkians and linear independence / Amer. Math. Monthly 117(8):722~727
L. Carlitz / 1940 / A set of polynomials / Duke Math. J. 6:486~504
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