A note on the use of R\'edei polynomials for solving the polynomial Pell equation and its generalization to higher degrees

The polynomial Pell equation is $$P^2-D{Q}^2=1$$ where $D$ is a given integer polynomial and the solutions $P,Q$ must be integer polynomials. A classical paper of Nathanson \cite{Nat} solved it when $D(x)=x^2+d$. We show that the R'edei polynomials can be used in a very simple and direct way for providing these solutions. Moreover, this approach allows to find all the integer polynomial solutions when $D(x)=f^2(x)+d$, for any $f \in \mathbb Z[X]$and$d \in \mathbb Z$, generalizing the result of Nathanson. We are also able to find solutions of some generalized polynomial Pell equations introducing an extension of R'edei polynomials to higher degrees.