## 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

### Keywords:

continued fraction expansion of infinite products

### Citations:

Zbl 0708.11010; Zbl 0749.11014
Full Text:

### 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
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