Continued fraction for the exponential for \(\mathbb{F}_ q[T]\)
type: Item
instance of: scholarly article
title: Continued fraction for the exponential for \(\mathbb{F}_ q[T]\)
zbMATH Open document ID: 0754.11019
DOI: 10.1016/0022-314X(92)90115-6
author: Dinesh S. Thakur
published in: Journal of Number Theory
publication date: 1993-01-16T00:00:00Z
review text: Let \(\mathbb{F}_ q\) be the finite field in \(q=p^ n\) elements. Put \({\mathbf A}=\mathbb{F}_ q[T]\). For \(i>0\) define \([i]:=T^{q^ i}-T\) and also the following sequence of elements of \({\mathbf A}:D_ 0:=1\), \(D_ j:=[j]D^ q_{j-1}\), \(j>0\). Basic to the arithmetic of \({\mathbf A}\) is the Carlitz exponential defined by \(e(z):=\sum^ \infty_{j=0}z^{q^ j}/D_ j\). This function is easily seen to be entire and additive; it uniformizes the Carlitz module which plays the role of \(\mathbb{G}_ m\) for \({\mathbf A}\). Thus the Carlitz exponential is a good analog of the classical exponential \(e^ z\). In the paper being reviewed, the author establishes a remarkable analog of the well-known continued fraction expansion of \(e^ z\) due to Euler. (We stress that the author's result is a genuine continued fraction expansion --- not just a generalized continued fraction expansion).
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 89646
zbMATH Keywords: Carlitz exponential
zbMATH Keywords: Carlitz module
zbMATH Keywords: continued fraction expansion
reviewed by: David Goss
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 89646