Parametric solutions of Pell equations

Abstract

This short paper is concerned with polynomial Pell equations \[P^2-DQ^2=1,\] with $P,Q,D\in \mathbb{C}[X]$ and $deg(D)=2$. The main result shows that the polynomials $P$ and $Q$ are closely related to Chebyshev polynomials. We then investigate the existence of such polynomials in $\mathbb{Z}[X]$ specializing to fixed solutions of ordinary Pell equations over the integers.