A generalized truncated logarithm

TL;DR

This paper introduces a generalized truncated logarithm function dependent on a parameter, which acts as an inverse to a parametrized truncated exponential related to Laguerre polynomials, with applications in algebraic grading techniques.

Contribution

It defines a new generalized truncated logarithm function and explores its properties as an inverse to a parametrized truncated exponential involving Laguerre polynomials, extending previous algebraic methods.

Findings

Defined a new generalized truncated logarithm $G^{(eta)}(X)$.

Established functional equations for $G^{(eta)}(X)$.

Connected the generalized logarithm to algebraic grading switching techniques.

Abstract

We introduce a generalization $G^{(\alpha)}(X)$ of the truncated logarithm $\mathcal{L}_1(X) = \sum_{k=1}^{p-1}X^k/k$ in 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 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 $\mathcal{L}_1(X)=-G^{(0)}(X)$.