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Karlin and McGregor's d-variable Hahn polynomials are shown to arise in the(d+1)-dimensional singular oscillator model as the overlap coefficients betweenbases associated to the separation of variables in Cartesian and hypersphericalcoordinates. These polynomials in d discrete variables depend on d+1 realparameters and are orthogonal with respect to the multidimensionalhypergeometric distribution. The focus is put on the d=2 case for which theconnection with the three-dimensional singular oscillator is used to derive themain properties of the polynomials: forward/backward shift operators,orthogonality relation, generating function, recurrence relations,bispectrality (difference equations) and explicit expression in terms of theunivariate Hahn polynomials. The extension of these results to an arbitrarynumber of variables is presented at the end of the paper.

The multivariate Hahn polynomials and the singular oscillator