## Mots infinis et produits de matrices à coefficients polynomiaux. (Infinite words and products of matrices the coefficients of which are polynomials).(French)Zbl 0758.11016

If $$(a_ n)_ n$$ is a $$q$$-automatic sequence, in the sense of G. Christol, T. Kamae, M. Mendès-France and G. Rauzy [Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)], with values in a field $$K$$, then the formal power series $$F(x)=\sum_{n=0}^ \infty a_ n x^ n$$ satisfies a Mahler equation, i.e. an equation $$a_ 0(x)F(x)+a_ 1F(x^ q)+a_ 2F(x^{q^ 2})+\dots+a_ d F(x^{q^ d})=0$$, where the $$a$$’s are polynomials in $$K[X]$$, not all zero.
The main result of this paper is a generalization of Mahler’s equations for sequences generated by substitutions which might have non constant length (a typical example being the Fibonacci sequence defined as the fixed point of the substitution $$1\to12$$, $$2\to1$$).

### MSC:

 11B85 Automata sequences 68Q45 Formal languages and automata 39B42 Matrix and operator functional equations

Zbl 0472.10035
Full Text:

### References:

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