Infinite products related to generalized Thue-Morse sequences

TL;DR

This paper investigates infinite products involving generalized Thue-Morse sequences, extending previous results on the classical Thue-Morse sequence, and explores their properties for rational functions.

Contribution

It introduces a framework for analyzing infinite products associated with generalized Thue-Morse sequences, broadening the scope of prior work on the classical sequence.

Findings

Derived explicit formulas for infinite products involving generalized Thue-Morse sequences.

Extended known results from the classical Thue-Morse sequence to a broader class of sequences.

Provided new insights into the structure and convergence of these infinite products.

Abstract

Given an integer $q\ge2$ and $\theta_1,\cdots,\theta_{q-1}\in\{0,1\}$, let $(\theta_n)_{n\ge0}$ be the generalized Thue-Morse sequence, defined to be the unique fixed point of the morphism $$0\mapsto0\theta_1\cdots\theta_{q-1}$$ $$1\mapsto1\overline{\theta}_1\cdots\overline{\theta}_{q-1}$$ beginning with $\theta_0:=0$, where $\overline{0}:=1$ and $\overline{1}:=0$. For rational functions $R$, we study infinite products of the forms $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{\theta_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{\theta_n}.$$ This generalizes relevant results given by Allouche, Riasat and Shallit in 2019 on infinite products related to the famous Thue-Morse sequence $(t_n)_{n\ge0}$ of the forms $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{t_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{t_n}.$$