## Substitution in two symbols and transcendence.(English)Zbl 0940.11015

The authors study substitutions in two symbols in connection with the dyadic expansion of real algebraic irrationals. Let $$A=\{a,b\}$$ be a set of two symbols, $$A^\ast$$ be the sets of all finite words over $$A$$ and $$\lambda$$ be the empty word. A substitution over $$A$$ is a map $$\sigma:A\to A^\ast\backslash\{\lambda\}$$. Let $$\Phi(X)=X^2-(t_{aa}+t_{bb})X +(t_{aa}t_{bb}-t_{ab}t_{ba})$$, where $$t_{\alpha\beta}=|\sigma(\alpha)|_{\beta}$$, the number of occurrences of the symbol $$\beta$$ in the word $$\sigma(\alpha)$$. Let $$w$$ be any fixed point of a substitution $$\sigma$$ in two symbols and let $$f_a(z)$$ and $$f_b(z)$$ be the generating functions of $$w$$ for $$a$$ and for $$b$$, respectively. If $$t_{ab}t_{ba}\Phi(0)\Phi(-1)\neq 0$$, then the numbers $$f_a(l^{-1})$$ and $$f_b(l^{-1})$$ are transcendental for any integer $$l\geq 2$$. In the case of substitutions in two symbols of constant length, the condition $$t_{ab}t_{ba}\Phi(0)\Phi(-1)\neq 0$$ can be removed. But in the case of nonconstant length this condition still remains.
Reviewer’s remark: S. Ferenczi and C. Mauduit [J. Number Theory 67, 146-161 (1997; Zbl 0895.11029)] and J.-P. Allouche and L. Q. Zamboni [J. Number Theory 69, 119-124 (1998; Zbl 0918.11016)] proved the same theorems by completely different methods.

### MSC:

 11B85 Automata sequences 11J81 Transcendence (general theory) 11A67 Other number representations

### Citations:

Zbl 0895.11029; Zbl 0918.11016
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