A generalized truncated logarithm

We introduce a generalization G(α)(X) of the truncated logarithm £1(X)=∑k=1p-1Xk/k in prime characteristic p, which depends on a parameter α. The main motivation of this study is G(α)(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 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(α)(X) motivated by known functional equations for £ 1(X) = - G(0)(X).