Definite sums and products over roots of polynomials

In Maple, the roots of are expressed as RootOf(a(x),x). Suppose alpha is the set of roots of the polynomial . Let sum(f(k),k=alpha) mean the sum of over the roots of . Likewise let product(f(k),k=alpha) be the product of over the roots of . Then Maple can evaluate such sums and products when is a polynomial or rational function in , e.g.

    > alpha := RootOf(x^5+x+11,x):
    > sum( 1/(k^2+1), k=alpha );
                                      12/125

    > product( k^2+1, k=alpha );
                                       125