2-Adic valuations of coefficients of certain integer powers of formal power series

摘要

Let epsilon = (epsilon(n))(n is an element of N) be an integer sequence and f(x) = Sigma(infinity)(n=0) epsilon(n)x(n) be its ordinary generating function. In this paper, we study the behavior of 2-adic valuations of the sequence (cm(n))(n is an element of N), where in m is an element of Z is fixed andf(x)(m) = Sigma(infinity)(n=0) c(m)(n)x(n).More precisely, we propose a method, which under suitable assumptions on the sequence epsilon allows us to prove boundedness of the sequence (nu(2)(c(m)(n)))(n is an element of N) for certain values of m is an element of Z. In particular, if epsilon is the classical Rudin-Shapiro sequence, then we prove that nu(2)(c(1-2s)(n)) is an element of {1,2,3} for given s is an element of N-= 2 and all n = 2(s). A similar result is proved for a relative of the Rudin Shapiro sequence recently introduced by Lafrance, Rampersad and Yee.