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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

Citations:

Zbl 0472.10035
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References:

[1] 1. J.-P. ALLOUCHE, Arithmétique et automates finis, Astérisque, 147-148, 1987, p. 13-26. Zbl0619.10005 MR891416 · Zbl 0619.10005
[2] 2. G. CHRISTOL, T. KAMAE, M. MENDÈS FRANCE et G. RAUZY, Suites algébriques, automates et substitutions, Bull. Soc. Math. France, 108, 1980, p. 164-192. Zbl0472.10035 MR614317 · Zbl 0472.10035
[3] 3. M. QUEFFÉLEC, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Math., 1294, 1987, Springer-Verlag. Zbl0642.28013 MR924156 · Zbl 0642.28013
[4] 4. Z.-X. WEN et Z.-Y. WEN, Sequences of Substitutions and Related Topics, Adv. in Math. China, 18, 1989, p. 270-293. Zbl0694.10006 MR1010491 · Zbl 0694.10006
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