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Transcendental numbers having explicit g-adic and Jacobi-Perron expansions. (English) Zbl 0763.11029
J. L. Davison proved in [Proc. Am. Math. Soc. 63, 29-32 (1977; Zbl 0326.10030)]: Let $$\alpha=(1+\sqrt 5)/2$$ and $$(f(j))$$, $$j\geq 0$$, be the sequence of Fibonacci numbers. Then $$x=x(2)=\sum^ \infty_{n=1} 2^{- [\alpha n]}$$ is transcendental and its continued fraction is given by $$a_ n(x)=2^{f(n)}$$. 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 $$\mathbb{Q}$$. (3) The Jacobi-Perron expansion can be described by the recurrence relation $$f_{n+3}=f_{n+2}+f_{n+1}+f_ n$$. An important tool in the proof (and of independent interest) is an associated Jacobi- Perron algorithm for formal Laurent series.

##### MSC:
 11J70 Continued fractions and generalizations 11J72 Irrationality; linear independence over a field 68Q45 Formal languages and automata
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##### References:
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