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

TL;DR

This paper constructs a family of continued fractions in the field of formal Laurent series over Q, using infinite words derived from the Thue-Morse sequence, and explicitly describes their continued fraction patterns.

Contribution

It introduces a new class of continued fractions associated with infinite binary words generated from the Thue-Morse sequence and generalizes to related sequences.

Findings

Explicit continued fraction patterns for each n > 1

Connection between Thue-Morse sequence and continued fractions

New elements in Q((T^-1)) with structured expansions

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.