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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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