## 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).

### MSC:

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

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