Power series approximations to Fekete polynomials

We study how well Fekete polynomials Fp(X)=∑n=0p−1npXn∈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.