×

Some explicit continued fraction expansions. (English) Zbl 0708.11011

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.1007/BF01746527 · Zbl 0179.02501
[2] Loxton, J. für Math. 392 pp 57– (1988)
[3] Blanchard, Bull. Sc. Math. 106 pp 325– (1982)
[4] DOI: 10.2307/1970953 · Zbl 0312.10024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.