Arithmetic properties of coefficients of power series expansion of ∏ n = 0 ∞ 1 - x 2 n t (with an appendix by Andrzej Schinzel)

• Published in: Monatshefte für Mathematik Vol. 185; no. 2; pp. 307 - 360
• Main Authors: Gawron, Maciej, Miska, Piotr, Ulas, Maciej
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.01.2018
• Subjects: Arithmetic; Convolution; Mathematical analysis; Power series; Series expansion; Thermal expansion
• ISSN: 0026-9255; 1436-5081
• DOI: 10.1007/s00605-017-1041-2

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(2k)=0, where k∈N, and fn(3)=0. Next, we present the exact value of the 2-adic valuation of the number fn(1-2m)—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.