On Hadamard Square Roots of Unity

A series all of whose coefficients have unit modulus is called an Hadamard square root of unity. We investigate and partially characterise the algebraic Hadamard square roots of unity

On Hadamard Square Roots of Unity

A series all of whose coefficients have unit modulus is called an Hadamard square root of unity. We investigate and partially characterise the algebraic Hadamard square roots of unity. The investigation makes use of a result about the asymptotic behavior of the coefficients of algebraic series and the Weyl-von Neumann theorem. Keywords: algebraic series, Hadamard product, analysis, formal power series 1 Introduction We work with power series over the complex numbers C . The Hadamard product h = f fi g of power series f and g is defined by [x n ]h = [x n ]f \Delta [x n ]g. The coefficient of x n in a series f is designated by [x n ]f . The identity element of the Hadamard product is the expansion of 1 1\Gammax . If f fi ¯ f = 1 1\Gammax , f is said to be an Hadamard square root of unity. Here, ¯ f is defined by [x n ] ¯ f = [x n ]f , where the bar indicates complex conjugation. It is evident that f is an Hadamard square root of unity iff all of its coefficients ha..