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 polynomialq n (z) given in monomial (!) representation can be evaluated for z ∈ 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. Classification (1991): 65D20, 42C10 Mathematics Subject

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