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

Roland Bacher (1)

1 IF - Institut Fourier

Using Chebyshev polynomials of 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 binomial coefficients which leads to a triangular array sharing many properties with Pascal's triangle.

Domaines: Combinatoire [math.CO]

Roland Bacher. Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle. 2015. ⟨hal-01206138⟩