A generalized truncated logarithm

Abstract

We introduce a generalization G (α) (X) of the truncated logarithm £ 1 (X) = p−1 k=1 X k /k in 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).

Generalized Finite Polylogarithms

“…1 (X ) is more compact than the original one we gave in [3,68 MARINA AVITABILE AND SANDRO MATTAREI Subsection 2.2]. Most of the work to bring that description to the fully factorized and arguably more useful form given here was actually done in [3,Section 4], with a short supplementary argument which we provide in Subsection 3.2 of this paper.…”

“…As explained there they can be viewed as special values of certain Jacobi polynomials, but what matters here are their full factorizations in F p [α], which were found in [3]. According to [3,Lemma 11], those polynomials satisfy…”

“…Those particular Laguerre polynomials were investigated by the authors in [2] as they play the role of generalized exponentials in a grading switching technique for modular, nonassociative algebras, whose purpose is to produce a new grading of an algebra from a given one. We limit ourselves here to giving the definition and exponential-like property of those Laguerre polynomials, referring the interested reader to a sketch of their role in grading switching in the Introduction of [3], and full details of that application in [2] and [1].…”

“…may be interpreted as a logarithm-like polynomial. Such inverse was investigated in the paper [3], where it was denoted by G (α) (X ). However, to match the standard notation £ 1 (X ) for the first finite polylogarithm, we set here £ (α) 1 (X ) = −G (α) (X ).…”

“…results from log exp(X ) = X upon viewing it first modulo X p and then modulo p. The details of this deduction are explained in the discussion following [3,Theorem 2].…”