# MANIPULATING POLYNOMIALS IN GENERALIZED FORM

We present new algorithms using structured
matrix methods for manipulating polynomials
expressed in generalized form,
that is, relative to a polynomial basis $\{p_i(x)\}$ 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.