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 $((-1{)}^{s_2(n)}{)}_{n\in \mathbb{N}}$. 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 $t\in {\mathbb{N}}_{+}$ the sequence $(f_n(t){)}_{n\in \mathbb{N}}$ is the Cauchy convolution of $t$ copies of the Prouhet-Thue-Morse sequence. For $t\in {\mathbb{Z}}_{<0}$ the $n$-th term of the sequence $(f_n(t){)}_{n\in \mathbb{N}}$ 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 $k\in \mathbb{N}$, 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.

arXiv comment34 pages, submitted