Sur le développement en fraction continue de la série de Baum et Sweet

Abstract

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On the algebraicity of Thue-Morse and period-doubling continued fractions

“…15 [13, 17] 1 [3,4] 6 [9,4] 11 [8,13] 16 [18, 4] 2 [5,6] 7 [10,4] 12 [14,4] 17 [19,12] 3 [1,7] 8 [11,6] 13 [15,16] 18 [16,6] 4 [4,4] 9 [6,12] 14 [12,16] 19 [17,8] and output function n → τ (n): For all k ≥ 1, a 2 2k−1 = a + 1. Therefore we only have to check that identity (3.7) holds for k = 3, 5, 7, which turns out to be true.…”

“…6 [11,12] 12 [15,20] 18 [23,12] 24 [27, 15] 1 [3,4] 7 [13,14] 13 [9,21] 19 [24, 23] 25 [15, 18] 2 [5,6] 8 [5,15] 14 [14,14] 20 [14,26] Let A(s, w) denote the state reached after reading w from right to left starting from the state s. For j = 0, 1, . .…”

“…As with real numbers, a formal Laurent series can also be represented by a continued fraction whose partial quotients are polynomials. Unlike for real numbers, the continued fraction expansion of an algebraic Laurent series of degree at least 3 may or may not have automatic partial quotients [4,5,14,2,15,11,12,13]; see also the introduction of [10].…”

A transcendence criterion in positive characteristic and applications

“…It was then generalized independently by L. Denis [6] and by Y. Hellegouarch [7] along two quite different lines. Inspired by the result of B. de Mathan and based on some considerations for automatic sequences, we gave in [15] (see also [14]) a series of transcendence criteria, and obtained, in particular, a new proof of the fact that the sequence of partial quotients of the formal power series of Baum-Sweet is not automatic, a result proved originally by M. Mkaouar in [9] by a quite different method.…”

On some conjectures of P. Barry

“…It would be interesting to find an example of a sequence b such that ϕ 1 (b) is not automatic. The examples we have in mind go the other way round: typically the Baum and Sweet sequence [6] is a 2-automatic sequence, but its associated power series has a continued fraction expansion -with partial quotients of degree 1 or 2-that is not automatic [33]. * As seen in Theorem 12 above, ϕ 2 (b) is 2-automatic if and only if b is 2-automatic.…”