  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.
