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

On a family of 2-automatic sequences generating algebraic continued fractions in characteristic 2

“…Here s(ε) is the celebrated period-doubling sequence. It has been proved that CF(s(ε)) satisfies an algebraic equation of degree 4 (see [7,8]). Each sequence in S is 2-automatic.…”

“…Continued fractions defined by two classical example of automatic sequences in characteristic 2 were studied in [7,5,8], where it was established that the Thue-Morse and period-doubling sequences both define algebraic continued fractions of degree 4. On the other hand, both sequences define quadratic formal power series.…”

Algebraic automatic continued fractions in characteristic $2$ II

“…In [3], the author and Han considered continued fractions defined by the Thue-Morse and the period-doubling sequence and proved their algebraicity in characteristic 2 for some cases. In [1], Bugeaud and Han proved the algebraicity of Thue-Morse continued fractions.…”

“…. is the Thue-Morse sequence t. Therefore one way to generalize [3], [1] and [4] is to consider continued fractions defined by σ n (p) for n ≥ 2.…”

The Thue–Morse continued fractions in characteristic 2 are algebraic

“…Conversely, very little is known about the Diophantine nature of a power series whose sequence of partial quotients is automatic, but not ultimately periodic. The only contribution to this question is a recent work of Hu and Han [19]. They proved that, for any distinct non-constant polynomials a and b in F 2 [z] whose sum of degrees is at most 7, the power series whose sequence of partial quotients is the Thue-Morse sequence written over {a, b} is algebraic of degree 4.…”