Arithmetic properties of coefficients of power series expansion of $$\prod _{n=0}^{\infty }\left( 1-x^{2^{n}}\right) ^{t}$$ ∏ n = 0 ∞

Abstract

(1 − x 2 n ) be the generating function for the ProuhetThue-Morse sequence ((−1) s 2 (n) ) n∈N . In this paper we initiate the study of the arithmetic properties of coefficients of the power series expansions of the functionFor t ∈ N + the sequence ( f n (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 ( f n (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 f n (2 k ) = 0, where k ∈ 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.

Values of Binary Partition Function Represented by a Sum of Three Squares

“…Indeed, b m (n) is the number of binary partitions of n, where each part has one of m possible colors. In a recent paper by Gawron et al [10], it is proved that for m = 2 k − 1 and n ≥ 2 k the 2-adic valuation of b m (n) belongs to the set {1, 2}. More precisely, they gave the following characterization of the 2-adic valuation of the terms b 2 k −1 (n).…”

“…More precisely, they gave the following characterization of the 2-adic valuation of the terms b 2 k −1 (n). Theorem 2.3 (Theorem 4.6 in [10]). Let k ∈ N + .…”

“…In particular, we have b(n) = b 1 (n). The study of arithmetic properties of this function was initiated in a paper of Gawron et al [10], where several useful results were obtained.…”

“…The reason for considering terms not represented as a sum of three squares is that the description turns out to be more concise (similarly as for non-negative integers). We focus primarily on the case m = 2 k − 1 for k ∈ N + due to the fact that the 2-adic valuation ν 2 (b 2 k −1 (n)) is known to be bounded [7,10]. This allows us to determine which terms b 2 k −1 (n) are of the form 4 r (8s+7) through the reduction modulo a suitable power of 2.…”

“…The result follows from the fact that a 2-regular sequence attaining finitely many values is necessarily 2-automatic.Unfortunately, we do not know even for a single value m = 2 k −1, whether or not the valuations ν 2 (b m (n)) are bounded. It is conjectured that they are unbounded for all m = 2 k −1 (see[10, Conjecture 5.3]). Nevertheless, this does not rule out 2-automaticity of the set S m .…”