Generalizations of Chebyshev polynomials and polynomial mappings

In this paper we show how polynomial mappings of degree $\mathfrak{K}$ from a union of disjoint intervals onto $[-1,1]$ generate a countable number of special cases of generalizations of Chebyshev polynomials. We also derive a new expression for these generalized Chebyshev polynomials for any genus $g$, from which the coefficients of $x^n$ can be found explicitly in terms of the branch points and the recurrence coefficients. We find that this representation is useful for specializing to polynomial mapping cases for small $\mathfrak{K}$ where we will have explicit expressions for the recurrence coefficients in terms of the branch points. We study in detail certain special cases of the polynomials for small degree mappings and prove a theorem concerning the location of the zeroes of the polynomials. We also derive an explicit expression for the discriminant for the genus 1 case of our Chebyshev polynomials that is valid for any configuration of the branch point.