Arithmetic properties of coefficients of power series expansion of $\prod_{n=0}^{\infty}\left(1-x^{2^{n}}\right)^{t}$ (with an Appendix by Andrzej Schinzel)

Abstract

Let $F(x)=\prod_{n=0}^{\infty}(1-x^{2^{n}})$ be the generating function for the Prouhet-Thue-Morse sequence $((-1)^{s_{2}(n)})_{n\in\N}$. In this paper we initiate the study of the arithmetic properties of coefficients of the power series expansions of the function $$ F_{t}(x)=F(x)^{t}=\sum_{n=0}^{\infty}f_{n}(t)x^{n}. $$ For $t\in\N_{+}$ the sequence $(f_{n}(t))_{n\in\N}$ is the Cauchy convolution of $t$ copies of the Prouhet-Thue-Morse sequence. For $t\in\Z_{<0}$ the $n$-th term of the sequence $(f_{n}(t))_{n\in\N}$ counts the number of representations of the number $n$ as a sum of powers of 2 where each summand can have one among $-t$ colors. Among other things, we present a characterization of the solutions of the equations $f_{n}(2^k)=0$, where $k\in\N$, and $f_{n}(3)=0$. Next, we present the exact value of the 2-adic valuation of the number $f_{n}(1-2^{m})$ - a result which generalizes the well known expression concerning the 2-adic valuation of the values of the binary partition function introduced by Euler and studied by Churchhouse and others.