Dualities in the q‐Askey Scheme and degenerate DAHA

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The Askey–Wilson polynomials are a four‐parameter family of orthogonal symmetric Laurent polynomials that are eigenfunctions of a second‐order q‐difference operator L, and of a second‐order difference operator in the variable n with eigenvalue (z+z^{-1}). Then, L and multiplication by (z+z^{-1}) generate the Askey–Wilson (Zhedanov) algebra. A nice property of the Askey–Wilson polynomials is that the variables z and n occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the nonsymmetric case and in the underlying algebraic structures: the Askey–Wilson algebra and the double affine Hecke algebra (DAHA). In this paper, we follow the degeneration of the Askey–Wilson polynomials until two arrows down and in four different situations: for the orthogonal polynomials themselves, for the degenerate Askey–Wilson algebras, for the nonsymmetric polynomials, and for the (degenerate) DAHA and its representations.


Original languageEnglish
Number of pages50
JournalStudies in Applied Mathematics
Early online date17 Sep 2018
Publication statusE-pub ahead of print - 17 Sep 2018