The paper gives a complete classification of A2, B2, and B2-prime contiguity relations for the finite Askey scheme families, and proves all A2 relations are Christoffel or Geronimus transforms.
A bivariate $Q$-polynomial structure for the non-binary Johnson scheme
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abstract
The notion of multivariate $P$- and $Q$-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. Numerous examples of such association schemes have already been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate $P$-polynomial association scheme. We show here that it is also a bivariate $Q$-polynomial association scheme for some parameters. This provides, with the $P$-polynomial structure, the bispectral property (i.e. the recurrence and difference relations) of a family of bivariate orthogonal polynomials made out of univariate Krawtchouk and dual Hahn polynomials. The algebra based on the bispectral operators is also studied together with the subconstituent algebra of this association scheme.
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Contiguity relations for finite families of orthogonal polynomials in the Askey scheme
The paper gives a complete classification of A2, B2, and B2-prime contiguity relations for the finite Askey scheme families, and proves all A2 relations are Christoffel or Geronimus transforms.