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)].


11A55 Continued fractions
11B83 Special sequences and polynomials
11A63 Radix representation; digital problems
Full Text: DOI


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