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

TL;DR

This paper proves that certain continued fractions over the field of formal Laurent series in characteristic 2, constructed from pairs of polynomials with partial quotients equal to either of the pair, are algebraic, extending previous results.

Contribution

The authors provide a short proof that all such continued fractions associated with the period-doubling sequence are algebraic over the field, generalizing earlier specific cases.

Findings

Continued fractions with partial quotients from a polynomial pair are algebraic in characteristic 2.

The proof applies to arbitrary pairs for the period-doubling sequence.

Extends previous results on algebraicity of continued fractions related to automatic sequences.

Abstract

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.