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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 Kettenbr├╝chen. 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

Citations:

JFM 55.0262.09
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Full Text: DOI

References:

[1] Perron, O, ()
[2] Hardy, G.H; Wright, E.M, ()
[3] Shrader-Frechette, M, ()
[4] Schneider, T, ()
[5] Khintchine, A.Ya, ()
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