Period-doubling Continued Fractions are Algebraic in Characteristic 2

For an arbitrary pair of distinct and non constant polynomials, a and b in 𝔽 2 [t], we build a continued fraction in 𝔽 2 ((1/t)) whose partial quotients are only equal to a or b. In a previous work of the first author and Han, the authors considered two cases where the sequence of partial quotients represents in each case a famous and basic 2-automatic sequence, both defined in a similar way by morphisms. They could prove the algebraicity of the corresponding continued fractions for several pairs (a,b) in the first case (the Prouhet–Thue–Morse sequence) and gave the proof for a particular pair for the second case (the period-doubling sequence). Recently Bugeaud and Han proved the algebraicity for an arbitrary pair in the first case. Here we give a short proof for an arbitrary pair in the second case.