On a family of continued fractions in $Q((T^1))$ associated to infinite binary words derived from the Thue-Morse sequence

TL;DR

The paper constructs a family of two-valued infinite words $W(i)$ derived from the Thue–Morse sequence and studies the associated generating functions $\theta_i=\sum w_{i,n} T^{-n}$ in $\mathbb{Q}((T^{-1}))$. It provides an explicit, computable description of the continued fraction expansions $\theta_i=[0,a_{i,1},a_{i,2},\dots]$ by decomposing partial quotients into leading coefficients $\lambda_{i,n}$ and unit polynomials $b_{i,n}$, with a simple periodic structure for $i=1$ and more intricate patterns for $i\ge2$ described via polynomials $P_{i,m}$. The work establishes growth and arithmetic properties, including irrationality measures $2i$ and transcendence of $\theta_i$ for $i\ge2$, supported by computational experiments and a referenced program. This demonstrates a broader class of infinite binary words whose associated continued fractions in $\mathbb{Q}((T^{-1}))$ can be explicitly characterized.

Abstract

For each integer n > 1, we present an element in $Q((T^-1))$, having a power series expansion based on an infinite word W(n), over the alphabet ${+1;-1}g and whose continued fraction expansion has a particular pattern which is explicitly described. The word W(1) is the Thue-Morse sequence and the following words are defined in a similar way.