## Simple continued fractions for some irrational numbers.(English)Zbl 0404.10003

Much is known about the continued fraction expansion of a real number $$x$$ and the connection between this expansion and algebraic properties of $$x$$ [see for instance: O. Perron, Die Lehre von den Kettenbrüchen. 2. Aufl. Leipzig: B. G. Teubner (1929; JFM 55.0262.09)]. Except for terminating and periodic expansions (rational $$x$$ and algebraic $$x$$ of degree 2 resp.) there are more well known patterns: in the expansion for $$e, e^2, \tanh(1/h), \exp(1/m), \exp(2/m)$$ etc.
In the paper under review the author derives definite information on the pattern in the expansion of the transcendental numbers $$B(u,\infty) = \sum_{k=0}^\infty u^{-2^k}$$ $$(u\ge 3$$, an integer). The proofs are elegant and only use well known properties of continued fractions and the principle of induction. Using $$B(u,v)$$ for the partial sums $$\sum_{k=0}^v$$ and $$[a_0, a_1, a_2,\ldots]$$ for the continued fraction $$a_0 + \frac{1 \vert}{\vert a_1} + \frac{1 \vert}{\vert a_2} + \cdots$$, the results are: $B(u,0) = [0,u],\quad B(u,1) = [0,u -1,u +1], \quad B(u,v)=[a_0, a_1, \ldots a_n] \tag{1} \Rightarrow$ 2. $$\quad$$ the expansion for $$B(u,v)$$ has $$2^v+1$$ partial denominators of which the first $$2^v$$ are identical to those in the expansion for $$B(u,\infty)$$.
$B(u,\infty) = [a_0, a_1, a_2,\ldots] \Rightarrow B(u+b,\infty) = [a_0, a_1+b, a_2+b,\ldots] \quad (u\ge 3,\ b\ge 0). \tag{3}$ 4. The mass (M. Shrader-Frechette) of $$B(u,v)$$ is $$u\cdot 2^v$$.
5. $$B(u,\infty)$$ has five unique partial denominators (thus bounded): $$0, u-2, u-1, u$$ and $$u+2$$. The number of occurrences of these numbers in the expansion for $$B(u,v)$$ is $$1, 2^{v2}- 1, 2, 2^{v-1}-1$$ and $$2^{v-2}$$, respectively.
6. If $$B(u,\infty) = [a_0, a_1,\ldots]$$ then $$a_n = u+2$$ if $$n\equiv 2\text{ or }7 \bmod 8$$ and $$= u$$ if $$n\equiv 3\text{ or }6 \bmod 8$$. Furthermore the numbers arising from adding a factor $$(-1)^k$$ in $$B(u,\infty)$$ are considered.

### MSC:

 11A55 Continued fractions 11J70 Continued fractions and generalizations

JFM 55.0262.09
Full Text:

### References:

 [1] Perron, O., (Die Lehre von den Kettenbrüchen, Vol. 1 (1957), Teubner: Teubner Stuttgart) · JFM 55.0262.09 [2] Hardy, G. H.; Wright, E. M., (An Introduction to the Theory of Numbers (1971), Clarendon: Clarendon Oxford) [3] Shrader-Frechette, M., (Continued Fractions, Farey Fractions, and Pascal’s Triangle. Continued Fractions, Farey Fractions, and Pascal’s Triangle, presented at the Miami University Number Theory Conference (September 30, 1977)) [4] Schneider, T., (Einführung in die Transzendenten Zahlen (1957), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0077.04703 [5] Khintchine, A. Ya, (Continued Fractions (1963), P. Noordhoff: P. Noordhoff Groningen) · Zbl 0117.28503
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