Manipulating Polynomials in Generalized Form

We present new algorithms using structured matrix methods for manipulating polynomi-als expressed in generalized form, that is, relative to a polynomial basis fp i (x)g satisfying a three-term recurrence relation. This includes computing the Euclidean remainders as well as the greatest common divisor of two polynomials u(x) and v(x) of degrees less than n. The computational schemes involve solving linear systems with nested Hankel matrices represented in the given generalized basis and they reach the optimal sequential complexity of O(n 2) arithmetical operations.