Generalizations of Chebyshev Polynomials and Polynomial Mappings

Abstract

We show how polynomial mappings of degree k from a union of disjoint intervals onto [-1,1] generate a countable number of special cases of a certain generalization of the Chebyshev Polynomials. We also derive a new expression for these generalized Chebyshev Polynomials for any number of disjoint intervals from which the coefficients of x^n can be found explicitly in terms of the end points and the recurrence coefficients. We find that this representation is useful for specializing to the polynomial mapping cases for small k where we will have algebraic expressions for the recurrence coefficients in terms of the end points. We study in detail certain special cases of the polynomials for small k and prove a theorem concerning the location of the zeroes of the polynomials. We also derive an expression for the discriminant for the case of two intervals that is valid for any configuration of the end points.