Continued fraction for the exponential for \(\mathbb{F}_ q[T]\). (English) Zbl 0754.11019

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).


11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
11J70 Continued fractions and generalizations
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