Abstract

We introduce a generalization $G^{(\alpha)}(X)$ of the truncated logarithm $\pounds_1(X)=\sum_{k=1}^{p-1}X^k/k$
in prime characteristic $p$, which depends on a parameter $\alpha$.
The main motivation of this study is $G^{(\alpha)}(X)$ being an inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential given by certain Laguerre polynomials.
Such Laguerre polynomials play a role in a {\em grading switching} technique for non-associative algebras,
previously developed by the authors, because they satisfy a weak analogue of the functional equation
$\exp(X)\exp(Y)=\exp(X+Y)$ of the exponential series.
We also investigate functional equations satisfied by $G^{(\alpha)}(X)$ motivated by known functional equations for $\pounds_1(X)=-G^{(0)}(X)$.