# A generalized truncated logarithm

Avitabile, Marina and Mattarei, Sandro (2019) A generalized truncated logarithm. Aequationes mathematicae, 93 (4). pp. 711-734. ISSN 0001-9054

Full content URL: http://doi.org/10.1007/s00010-018-0608-x

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## 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 $\pounds_1(X)=-G^{(0)}(X)$.

Additional Information: The final published version can be accessed online at https://link.springer.com/journal/10 truncated logarithm; polylogarithm; Laguerre polynomial; functional equation; Jacobi polynomial G Mathematical and Computer Sciences > G110 Pure Mathematics College of Science 33203 13 Sep 2018 14:32

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