## Folded continued fractions.(English)Zbl 0753.11005

This paper exhibits a class of Laurent series for which the continued fraction expansions can be displayed from a simple algorithm. In order to give a significant result, let us consider the field $$L$$ of formal Laurent series in the variable $$X^{-1}$$ with real coefficients, endowed with the usual valuation at infinity. Notation $$[a_ 0,a_ 1,\ldots,a_ n]$$ $$(a_ j\in L)$$ is defined recursively by $$[a_ 0]=a_ 0$$ and $$[a_ 0,a_ 1,\ldots,a_ n]=a_ 0+[a_ 1,\ldots,a_ n]^{-1}$$ if the inverse exists. If the sequence $$n\mapsto[a_ 0,a_ 1,\ldots,a_ n]$$ converges in $$L$$ the limit is denoted by $$[a_ 0,a_ 1,\ldots]$$. Continued fraction expansion $$F(X)=[a_ 0,a_ 1,a_ 2,\ldots]$$ of a formal Laurent series $$F(X)$$ requires the partial quotients $$a_ n$$ to be polynomials in $$X$$ of degree at least 1, except $$a_ 0$$, which may be any polynomial. For any string of polynomial $$w=(w_ 1,\ldots,w_ s)$$ let $${\mathcal F}_ w$$ be the so-called folding operator defined by ${\mathcal F}_ w([\alpha_ 1,\ldots,\alpha_ m])=[w_ 1,\ldots,w_ s,\alpha_ 1,\ldots,\alpha_ m, -w_ s,\ldots,-w_ 1].$ where the $$\alpha_ i$$ are polynomials. This operator is formally connected with paperfolding sequences [see the survey of M. Dekking, M. Mendès-France and A. J. van der Poorten, Math. Intell. 4, 130-138, 173-181, 190-195 (1982; Zbl 0493.10001-10003)]. The authors prove (Theorem 1): Let $$F(X)=X(X^{- 1}+\sum^ \infty_{h=1}j_ hX^{-2^ h})$$ be a Laurent series with $$j_ h=\pm 1$$. Then the following continued fraction expansion holds: $F(X)=[1,\ldots,{\mathcal F}_{j_ 4X}{\mathcal F}_{j_ 3X}{\mathcal F}_{j_ 2X}([j_ 1X])].$ Putting $$X=2$$ leads to the irregular continued fraction expansion $$F(2)=[1,2f_ 1,2f_ 2,2f_ 3,\dots]$$ where $$(f_ n)_ n$$ is a $$\pm 1$$-paperfolding sequence. Finally, the property $$f_{2h+1}=(-1)^ hf_ 1$$ is used to prove (Theorem 2) that nonnegative real numbers of the form $$2\sum^ \infty_{h=0}\pm 2^{- 2^ h}$$ all have continued fraction expansions with partial quotients 1 or 2. The paper contains other interesting results which can be used to rediscover previous results, e.g. the continued fraction expansions of the so-called Fredholm numbers $$\sum^ \infty_{h=0}\pm g^{-2^ h}$$, obtained by M. Kmosek or J. Shallit [J. Number Theory 11, 209-217 (1979; Zbl 0404.10003), Fibonacci Q. 20, 77-81 (1982; Zbl 0472.10012)].

### MSC:

 11A55 Continued fractions 11B83 Special sequences and polynomials 11A63 Radix representation; digital problems
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### References:

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