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.