Arithmetic properties of coefficients of power series expansion of ∏n=0∞(1-x2n)t (with an appendix by Andrzej Schinzel)

Abstract

Let F(x)=∏n=0∞(1-x2n) be the generating function for the Prouhet–Thue–Morse sequence ((-1)s2(n))n∈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.For t∈ N+ the sequence (fn(t))n∈N is the Cauchy convolution of t copies of the Prouhet–Thue–Morse sequence. For t∈ Z< 0 the n-th term of the sequence (fn(t))n∈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 fn(2 k) = 0 , where k∈ N, and fn(3 ) = 0. Next, we present the exact value of the 2-adic valuation of the number fn(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.