  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.
