A lower bound for the discriminant of polynomials related to Chebyshev polynomials

Abstract

In this paper we give a lower bound for the discriminant of the polynomials {Gk,i(x)} defined by Gk,0(x)=1,Gk,1(x)=x+1 and Gk,i+2(x)=xGk,i+1(x)−(k−1)Gk,i(x)fori≥0. These polynomials are closely related to the Chebyshev polynomials of the second kind. We derive our result by using orthogonality properties of the polynomials {Fk,i(x)} defined by Fk,0(x)=1,Fk,1(x)=x,Fk,2(x)=x2−k and Fk,i+2(x)=xFk,i+1(x)−(k−1)Fk,i(x)fori≥1.