On algebraic properties of classical multiple orthogonal polynomials of discrete variable

In this talk, we shall consider the class of multiple orthogonal polynomials with respect to $d$ discrete measures. These measures are supported of the shifts of integer lattices and the weight functions are the product of the classical weights by Charlier, Meixner, Kravchuk (i.e. Krawtchouk) and Hahn. Recently we found a difference analogue of the Rodrigues formula for these polynomials and, for $d=2$, fully classified these polynomials in the case when the measures are positive. These polynomials turn to keep certain algebraic properties of classical discrete orthogonal polynomials: in particular, we derived the corresponding step-line recurrence relations and third-order linear difference equation. Our motivation stems from certain applications: for example, for the marginal indices (if one of the indices is zero – so only one of the orthogonality measures remains) the recurrence coefficients satisfy various Painlevé equations.