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

TL;DR

This paper explores the use of Chebyshev polynomials to generate rational approximations of quadratic surds and introduces a new triangular array with properties similar to Pascal's triangle.

Contribution

It presents novel identities involving Chebyshev polynomials and binomial coefficients, leading to a Pascal-like triangular array with unique properties.

Findings

Constructed rational fractions as convergents of quadratic surds

Discovered identities linking Chebyshev polynomials and binomial coefficients

Introduced a Pascal-like triangular array with distinctive properties

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.