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Period-doubling Continued Fractions are Algebraic in Characteristic $2$

Considering an arbitrary pair of distinct and non constant polynomials, $a$ and $b$ in $\mathbb{F}_2[t]$, we build a continued fraction in $\mathbb{F}_2((1/t))$ whose partial quotients are only equal to $a$ or $b$. In a previous work of the first author and Han (to appear in Acta Arithmetica), 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 (arXiv:2203.02213) 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.