Note on the Stern-Brocot sequence, some relatives, and their generating power series
Peter Bundschuh1; Keijo Väänänen2
1 Mathematisches Institut Universität zu Köln Weyertal 86–90 50931 Köln, Germany
2 Department of Mathematical Sciences University of Oulu P. O. Box 3000 90014 Oulu, Finland
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 195-202.

Three variations on the Stern-Brocot sequence are related to the celebrated Thue-Morse sequence. In the present note, the generating power series of these four sequences are considered. Whereas one of these was known to define a rational function, the other three are proved here to be algebraically independent over ℂ(z). Then this statement is fairly generalized by including the functions Φ(z),Φ(-z),Ψ(z),Ψ(z 2 ), where Φ and Ψ denote the generating power series of the Rudin-Shapiro and Baum-Sweet sequence, respectively. Moreover, some arithmetical applications are given.

Trois variantes de la suite de Stern-Brocot sont liées à la célèbre suite de Thue-Morse. Dans la présente note, les fonctions génératrices de ces quatre suites sont considérées. Tandis que l’une d’entre elles est connue comme étant rationnelle, l’indépendance algébrique sur ℂ(z) des trois autres est démontrée ici. Puis, ce théorème est généralisé de sorte que les fonctions Φ(z),Φ(-z),Ψ(z),Ψ(z 2 ) sont aussi considérées, où Φ et Ψ indiquent les fonctions génératrices des suites de Rudin-Shapiro et de Baum-Sweet, respectivement. Quelques applications arithmétiques sont également données.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1022
Classification: 11J81, 11J91, 11B37
Mots-clés : Stern-Brocot sequence, transcendence, algebraic independence, Mahler’s method

Peter Bundschuh 1; Keijo Väänänen 2

1 Mathematisches Institut Universität zu Köln Weyertal 86–90 50931 Köln, Germany
2 Department of Mathematical Sciences University of Oulu P. O. Box 3000 90014 Oulu, Finland
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Peter Bundschuh; Keijo Väänänen. Note on the Stern-Brocot sequence, some relatives, and their generating power series. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 195-202. doi : 10.5802/jtnb.1022. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1022/

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