### Bispectral commuting partial dierence operators for multivariate hypergeometric polynomials

The classical discrete orthogonal polynomials can be characterized by the fact that they are eigenfunctions

of a second-order dierence 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 specic basis) of orthogonal polynomials. I will discuss

recent results related to the construction and the classication of multivariate orthogonal polynomials which

are eigenfunctions of two bispectral commutative algebras of partial dierence operators: one acting on

the variables of the polynomials and the other one on their degree indices.

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