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The Askey–Wilson polynomials are a four‐parameter family of orthogonal symmetric Laurent polynomialsR  n    [  z  ]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    =  2  x   . Then,  L  and multiplication by    z  +   z   −  1generate 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.

Dualities in the q ‐Askey Scheme and Degenerate DAHA