Proof of the Gawron-Miska-Ulas conjecture concerning unboundedness of coefficients of power series expansion of $\prod_{n=0}^{\infty}(1-x^{2^{n}})^m$

Abstract

It is well known that $F(x)=\prod_{n=0}^{\infty}(1-x^{2^n})$ is the generating function of the Prouhet-Thue-Morse sequence $\{(-1)^{σ_2(n)}\}_{n=0}^\infty$, where $σ_2(n)$ is the sum of (binary) digits of $n$. Let $m$ be an integer. In 2018, Gawron, Miska and Ulas initiated the study of arithmetic properties of power series expansion of the function $$F_m(x)=F(x)^m=\sum_{n=0}^{\infty}t_m(n) x^n,$$ and proposed a conjecture stating that for any given integer $m\ge 2$, the sequence $\{t_m(n)\}_{n=0}^{\infty}$ is unbounded. In this paper, we introduce a new method to investigate this conjecture. In fact, by making use of algebraic, $p$-adic and analytic methods, we show that the Gawron-Miska-Ulas conjecture is true.