Fq-Linear Calculus over Function Fields Original Research Article

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