Power Series Approximations to Fekete Polynomials

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.