Parametric solutions of Pell equations

TL;DR

This paper explores polynomial Pell equations with quadratic polynomials, revealing their connection to Chebyshev polynomials and examining integer solutions that specialize to classical Pell solutions.

Contribution

It establishes a link between polynomial Pell solutions and Chebyshev polynomials, and investigates integer solutions related to classical Pell equations.

Findings

Polynomials P and Q are related to Chebyshev polynomials.

Conditions for integer solutions of polynomial Pell equations.

Existence criteria for specialized solutions over integers.

Abstract

This short paper is concerned with polynomial Pell equations \[P^2-DQ^2=1,\] with $P,Q,D\in\Bbb C[X]$ 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 $\Bbb Z[X]$ specializing to fixed solutions of ordinary Pell equations over the integers.