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)})_{nin {mathbb {N}}}. In this paper we initiate the study of the arithmetic properties of coefficients of the power series expansions of the function Ft(x)=F(x)t=∑n=0∞fn(t)xn.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} F_{t}(x)=F(x)^{t}=\\sum _{n=0}^{\\infty }f_{n}(t)x^{n}. \\end{aligned}$$\\end{document}For tin {mathbb {N}}_{+} the sequence (f_{n}(t))_{nin {mathbb {N}}} is the Cauchy convolution of t copies of the Prouhet–Thue–Morse sequence. For tin {mathbb {Z}}_{<0} the n-th term of the sequence (f_{n}(t))_{nin {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 kin {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.