Fast associated classical orthogonal polynomial transforms

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