Some explicit continued fraction expansions. (English) Zbl 0708.11011

Mendès France, Michel; van der Poorten, Alfred J.

Mathematika 38, No. 1, 1-9 (1991).

The authors discuss the continued fraction expansion of infinite products \(\prod^{\infty}_{h=0}(1+x^{-\lambda_ h})\) which are elements of a field \(K((X^{-1}))\) where \(K\) is any field. The continued fraction is completely explicit provided the \(\lambda_ h\) are increasing integers such that \(\lambda_{h+1}/2\lambda_ h\) all are integers larger than or equal to 2. The analysis applies in particular when \(\lambda_ h=a^ h\) where \(a\) is an even integer.
In a forthcoming paper [An infinite product with bounded partial quotients (to appear in Acta Arith. 59, No. 2, 171–182 (1991; Zbl 0749.11014)], J. P. Allouche and the authors show that the case where \(a\) is an odd integer is altogether different. Suppose \(K={\mathbb R}\). If \(c_ n(X,a)\) is the \(n\)th partial quotient of \(\prod^{\infty}_{h=0}(1+X^{-a^ h})\) then \[ \liminf_{n\to \infty}\deg c_ n(X,a)=\infty \quad if\quad a\equiv 0\pmod 2;\quad =1\quad if\quad a\equiv 1\pmod 2. \]

Reviewer: M.Mendès France

MSC:

11A55	Continued fractions
11J70	Continued fractions and generalizations

Keywords:

continued fraction expansion of infinite products

Citations:

Zbl 0708.11010; Zbl 0749.11014

References:

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[3]	Blanchard, Bull. Sc. Math. 106 pp 325– (1982)
[4]	DOI: 10.2307/1970953 · Zbl 0312.10024