A note on the maximal coefficients of squares of Newman polynomials
A Newman polynomial [\textit{D. J. Newman}, Proc. Am. Math. Soc. 16, 1287--1290 (1965; Zbl 0151.08103)] is a polynomial \(p_i\) (\(i\) the degree of \(p_i\)) with coefficients in the set \(\{0,1\}\). With \((\# p_i)\) is denoted the number of non-zero coefficients of the Newman polynomial \(p_i\) and \(M(p^2)i)\) is the maximal coefficient of \(p^2_i\). The main result of this paper is the following theorem: For \(n= 4k\), is given a family \(G= \{g_n\}\) of Newman polynomials of degree \(n\) via the identity \[ xg_n(x):= \sum^{n/4}_{m=1} x^m+ \sum^n_{m= n/2} x^{m+1}. \] Then \(\lim_{k\to\infty}{\deg(g_{4k})M(g^2_{4k})\over (\neq g_{4k})^2}= {8\over 9}\). This investigation is suggested by a recent paper of \textit{Gang Yu} [J. Number Theory 122, No. 1, 211--220 (2007; Zbl 1201.11035)]. Finally are given some open questions.
scientific article; zbMATH DE number 5190743