Sobolewski, Bartosz Cyclotomic properties of polynomials associated with automatic sequences. (English) Zbl 1480.11031 Adv. Appl. Math. 120, Article ID 102063, 24 p. (2020). 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 \(\pm 1\). Reviewer: Jeffrey Shallit (Waterloo) MSC: 11B85 Automata sequences 11B37 Recurrences 11C08 Polynomials in number theory 11R18 Cyclotomic extensions 68Q45 Formal languages and automata Keywords:automatic sequences; polynomials; cyclotomy; recurrence relations Citations:Zbl 0335.12003 Software:Mathematica PDFBibTeX XMLCite \textit{B. Sobolewski}, Adv. Appl. Math. 120, Article ID 102063, 24 p. (2020; Zbl 1480.11031) Full Text: DOI arXiv References: [1] Allouche, Jean-Paul; Shallit, Jeffrey, Automatic Sequences. Theory, Applications, Generalizations (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1086.11015 [2] Brillhart, John; Lomont, John S.; Morton, Patrick, Cyclotomic properties of the Rudin-Shapiro polynomials, J. Reine Angew. Math., 288, 37-65 (1976) · Zbl 0335.12003 [3] Doche, Christophe, On the real roots of generalized Thue-Morse polynomials, Acta Arith., 99, 4, 309-319 (2001) · Zbl 1020.11017 [4] Doche, Christophe; Mendès France, Michel, Integral geometry and real zeros of Thue-Morse polynomials, Exp. Math., 9, 3, 339-350 (2000) · Zbl 0976.52005 [5] Kozen, Dexter C., Automata and Computability (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0883.68055 [6] Rudin, Walter, Some theorems on Fourier coefficients, Proc. Am. Math. Soc., 10, 855-859 (1959) · Zbl 0091.05706 [7] Shapiro, Harold S., Extremal problems for polynomials and power series (1951), Master’s Thesis, M.I.T., Retrieved from [8] Tijsma, Djurre J., Gaussian periods (2016), Bachelor’s Thesis, Utrecht University, Retrieved from [9] Research Wolfram Inc., Mathematica, Version 11.2 (2017), Champaign, IL This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.