Two basic discrete Darboux transformations in the theory of orthogonal polynomials are called Geronimus and Christoffel transformations. The consistency relation for those two gives the discrete Toda equation, a discrete integrable system, and it can also be considered as a relation between the elements of the Padé table.

In this talk, we are going to review the basics of discrete Darboux transformations for orthogonal polynomials. Then we'll show how such transformations can lead to Sobolev orthogonal polynomials, exceptional orthogonal polynomials, and indefinite orthogonal polynomials. Some associated asymptotic results for orthogonal polynomials and convergence results for underlying Padé approximants will be presented as well. Speaker(s): Maxim Derevyagin (University of Connecticut)