Arithmetic properties of coefficients of power series expansion of $$\prod _{n=0}^{\infty }\left( 1-x^{2^{n}}\right) ^{t}$$ ∏ n = 0 ∞ 1 - x 2 n t (with an appendix by Andrzej Schinzel)
Maciej Gawron, Piotr Miska, Maciej Ulas · Monatshefte für Mathematik · 2017
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 $$\begin{aligned} F_{t}(x)=F(x)^{t}=\sum _{n=0}^{\infty }f_{n}(t)x^{n}. \end{aligned}$$ 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.