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 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 \(\pounds _1(X)=-G^{(0)}(X)\).
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Avitabile, M., Mattarei, S. A generalized truncated logarithm. Aequat. Math. 93, 711–734 (2019). https://doi.org/10.1007/s00010-018-0608-x
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DOI: https://doi.org/10.1007/s00010-018-0608-x