Donetsk Institute for Physics and Technology
Donetsk 83114, Ukraine
E-mail: zhedanov@kinetic.ac.donetsk.ua

Two-dimensional Krall-Sheffer orthogonal polynomials and integrable systems

Abstract:
In 60-s Krall and Sheffer classified all orthogonal polynomials in two variables which are eigensolutions of a second-order partial differential operator (with some natural restrictions). The  Krall-Sheffer polynomials can be considered as a two-variable extension  of the well-known classical one-variable orthogonal polynomials (i.e. Jacobi, Laguerre, Hermite and Bessel polynomials). Krall and Sheffer showed that there are essentially 9 distinct types of such polynomials in two variables. We show that all 9 types of the Krall-Sheffer polynomials are connected with integrable systems on spaces with constant curvature.