2017/03/06 by Maciej Gawron, Gawron, Maciej, Piotr Miska +3
Mathematics · #11B50 #11P81 #11P83 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1703.01955
openalex publication_date 2017/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F(x)=∏n=0∞(1-x^2n) 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.