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Cyclotomic properties of polynomials associated with automatic sequences. (English) Zbl 1480.11031

Let \((a_n)_{n \geq 0}\) be a sequence, and define \(T_n (X) = \sum_{0 \leq i < n} a_i X^i\). Brillhart, Lomont, and Morton [J. Brillhart et al., J. Reine Angew. Math. 288, 37–65 (1976; Zbl 0335.12003)] showed that if \((a_n)\) is the Rudin-Shapiro sequence with values \(\pm 1\), and \(\zeta\) is an \(r\)-th root of unity, where \(r>1\) is odd, then \[ T_{n+2s} (\zeta) - C_s(\zeta) T_{n+s} (\zeta) + (-2)^s T_n(\zeta) = 0 \] for a certain sequence of polynomials \(C_s (X)\). They also investigated when \(C_s(\zeta) \in \mathbb{Z}\).
In this paper the author generalizes the result of Brillhart, Lomont, and Morton to arbitrary \(k\)-automatic sequences; these are sequences \((a_n)\) computed by finite automata that read an input \(n\) expressed in base \(k\) and produce an output associated with the last state reached. He investigates when the resulting polynomial coefficients take on integer values at roots of unity. He also obtains detailed results in the case that \((a_n)\) is the Thue-Morse sequence with values in \(\pm 1\).

MSC:

11B85 Automata sequences
11B37 Recurrences
11C08 Polynomials in number theory
11R18 Cyclotomic extensions
68Q45 Formal languages and automata

Citations:

Zbl 0335.12003

Software:

Mathematica
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Full Text: DOI arXiv

References:

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