hBLinear Calculus over Function Fields, by Anatoly N. Kochubei

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.

Anatoly N. Kochubei <ank@ank.kiev.ua>