Bispectral commuting partial di erence operators for multivariate hypergeometric polynomials
The classical discrete orthogonal polynomials can be characterized by the fact that they are eigenfunctions
of a second-order di erence operator. For some families, this operator can be connected to the three-term
recurrence relation via a duality between the variable and the degree index of the polynomials.
One of the main obstacles for the extension of the above theory to higher dimensions is the fact that
the polynomials are no longer uniquely determined by the orthogonality measure (up to a multiplicative
constant). Thus, the recurrence relations, the spectral properties and the duality between the variables and
the degree indices depend on the construction (the speci c basis) of orthogonal polynomials. I will discuss
recent results related to the construction and the classi cation of multivariate orthogonal polynomials which
are eigenfunctions of two bispectral commutative algebras of partial di erence operators: one acting on
the variables of the polynomials and the other one on their degree indices.
