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Zbl 0763.11029
Tamura, Jun-ichi
Transcendental numbers having explicit g-adic and Jacobi-Perron expansions. (English)
[J] Sémin. Théor. Nombres Bordx., Sér. II 4, No.1, 75-95 (1992). ISSN 0989-5558

{\it J. L. Davison} proved in [Proc. Am. Math. Soc. 63, 29-32 (1977; Zbl 0326.10030)]: Let $\alpha=(1+\sqrt 5)/2$ and $(f(j))$, $j\ge 0$, be the sequence of Fibonacci numbers. Then $x=x(2)=\sum\sp \infty\sb{n=1} 2\sp{- [\alpha n]}$ is transcendental and its continued fraction is given by $a\sb n(x)=2\sp{f(n)}$. The binary expansion of $x$ can be described as the fixed point of a substitution over a finite alphabet. In this paper a similar theorem is proved for pairs of numbers $\bigl(x(g),y(g)\bigr)$: (1) Their $g$-adic expansions are given by fixed points of a substitution. (2) The numbers 1, $x(g)$, $y(g)$ are linearly independent over $\bbfQ$. (3) The Jacobi-Perron expansion can be described by the recurrence relation $f\sb{n+3}=f\sb{n+2}+f\sb{n+1}+f\sb n$. An important tool in the proof (and of independent interest) is an associated Jacobi- Perron algorithm for formal Laurent series.
[F.Schweiger (Salzburg)]

MSC 2000:: *11J70 Continued fractions and generalizations
11J72 Irrationality
68Q45 Formal languages

Keywords: irrationality; $g$-adic expansions; fixed points; substitution; Jacobi- Perron expansion; associated Jacobi-Perron algorithm for formal Laurent series

Citations: Zbl 0347.10028; Zbl 0326.10030

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