On a two-valued sequence and related continued fractions in power series fields

Bill Allombert (1, 2) , Nicolas Brisebarre (3, 4) , Alain Lasjaunias (1)

1 IMB - Institut de Mathématiques de Bordeaux
2 LFANT - Lithe and fast algorithmic number theory
3 ARIC - Arithmetic and Computing
4 LIP - Laboratoire de l'Informatique du Parallélisme

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set {1, 2}. The origin of this sequence, whose study was initiated in a recent paper, is to be found in another continued fraction, in the field of power series over $\mathbb{F}_3$, which satisfies a simple algebraic equation of degree 4, introduced thirty years ago by D. Robbins.

Mots clés: power series over a finite field; finite alphabet; finite automata; formal power series; continued fractions; words; automatic sequences

Domaines: Théorie des nombres [math.NT]

Bill Allombert, Nicolas Brisebarre, Alain Lasjaunias. On a two-valued sequence and related continued fractions in power series fields. Ramanujan Journal, 2018, 45 (3), pp.859-871. ⟨10.1007/s11139-017-9892-7⟩. ⟨hal-01348576v3⟩