Editing Chebyshev polynomials (section)

===Pell equation definition===
The Chebyshev polynomials can also be defined as the solutions to the [[Pell equation]]:
<math display="block">T_n(x)^2 - \left(x^2 - 1\right) U_{n-1}(x)^2 = 1</math>
in a [[ring (mathematics)|ring]] {{math|''R''[''x'']}}.<ref>{{cite thesis |first=Jeroen |last=Demeyer |url=http://cage.ugent.be/~jdemeyer/phd.pdf |title=Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields |archive-url=https://web.archive.org/web/20070702185523/https://cage.ugent.be/~jdemeyer/phd.pdf |archive-date=2007-07-02 |degree=Ph.D. |year=2007 |page=70}}</ref> Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:
<math display="block">T_n(x) + U_{n-1}(x)\,\sqrt{x^2-1} = \left(x + \sqrt{x^2-1}\right)^n~. </math>