Power Series Approximations to Fekete Polynomials

Abstract

We study how well Fekete polynomials $$F_p(X)=\sum _{n=0}^{p-1}\left(\frac{n}{p}\right)X^n\in \mathbb{Z}[X]$$ with the coefficients given by Legendre symbols modulo a prime $p$, can be approximated by power series representing algebraic functions of a given degree. We also obtain some explicit results describing polynomial recurrence relations which are satisfied by the coefficients of such algebraic functions.