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 $\{(-1{)}^{{\sigma }_2(n)}{\}}_{n=0}^{\infty }$, where ${\sigma }_2(n)$ 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.

21 pages