Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle

Abstract

Using Chebyshev polynomialsof both kinds, we construct rational fractions which are convergents of the smallest root of $x^2-\alpha x+1$ for $\alpha=3,4,5,\dots$.Some of the underlying identities suggest an identity involving binomialcoefficients which leads to a triangular array sharing many propertieswith Pascal's triangle.