On orthogonal polynomials spanning a non-standard flag

Understand

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations.

• The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials.
• Unlike the classical families, these new examples, called exceptional orthogonal polynomials, feature non-standard polynomial flags; the lowest degree polynomial has degree $m>0$.
• In this paper we review the classification of codimension $m=1$ exceptional polynomials, and give a novel, compact proof of the fundamental classification theorem for codimension 1 polynomial flags.