Polynomial evaluation and associated polynomials

Abstract.

We show a connection between the Clenshaw algorithm for evaluating a polynomial $q_n$, expanded in terms of a system of orthogonal polynomials, and special linear combinations of associated polynomials. These results enable us to get the derivatives of $q_n(z)$ analogously to the Horner algorithm for evaluating polynomials in monomial representations. Furthermore we show how a polynomial $\hat q_n(z)$ given in monomial (!) representation can be evaluated for $z\in{\Bbb C}$ using the Clenshaw algorithm without complex arithmetic. From this we get a connection between zeros of polynomials expanded in terms of Chebyshev polynomials and the corresponding polynomials in monomial representation with the same coefficients.