Discrete multiple orthogonal polynomials on shifted lattices

There are many ways to define multiple orthogonal polynomials with respect to the classical continuous weights. Bearing in mind a deep connection between the classical discrete and continuous orthogonality, we adapt to the discrete case the approach as in [1-3] preserving a kind of the Rodrigues formula. Our work [4] introduced a new class of polynomials of multiple orthogonality with respect to the product of classical discrete weights on integer lattices with noninteger shifts.

This talk is devoted to further progress in this direction for the case of two measures. In particular, we obtain coefficients for the four-term recurrence relations connecting polynomials with indices on “diagonals” (including the “step line”). The initial conditions for these relations are presented by semi-classical extensions of discrete orthogonal polynomials studied in [5-7].

This is a joint work with Vladimir Lysov.