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$F_q$-Linear Calculus over Function Fields

Anatoly N. Kochubei
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Abstract

We define analogues of higher derivatives for $F_q$-linear functions over the field of formal Laurent series with coefficients in $F_q$. This results in a formula for Taylor coefficients of a $F_q$-linear holomorphic function, a definition of classes of $F_q$-linear smooth functions which are characterized in terms of coefficients of their Fourier-Carlitz expansions. A Volkenborn-type integration theory for $F_q$-linear functions is developed; in particular, an integral representation of the Carlitz logarithm is obtained.

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Anatoly N. Kochubei. 1998-07-15. $F_q$-Linear Calculus over Function Fields. https://arxiv.org/abs/math/9807075

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