  We discuss a fast approximate solution to the associated classical --
classical orthogonal polynomial connection problem. We first show that
associated classical orthogonal polynomials are solutions to a fourth-order
quadratic eigenvalue problem with polynomial coefficients such that the
differential operator is degree-preserving. Upon linearization, the
discretization of this quadratic eigenvalue problem is block upper-triangular
and banded. After a perfect shuffle, we extend a divide-and-conquer approach to
the upper-triangular and banded generalized eigenvalue problem to the blocked
case, which may be accelerated by one of a few different algorithms. Associated
orthogonal polynomials arise from iterated Stieltjes transforms of orthogonal
polynomials; hence, fast approximate conversion to classical cases combined
with fast discrete sine and cosine transforms provides a modular mechanism for
synthesis of singular integral transforms of classical orthogonal polynomial
expansions.
