Dualities in the q‐Askey Scheme and Degenerate DAHA. Issue 4 (17th September 2018)

• Record Type: Journal Article
• Authors: Koornwinder, Tom H.; Mazzocco, Marta
• Other Names: Clarkson Peter guestEditor.; Daalhuis Adri Olde guestEditor.
• Is Part Of: Studies in applied mathematics. Volume 141:Issue 4(2018)
• Page Start: 424
• Page End: 473
• Publication Date: 2018-09-17
• Subjects: Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12229
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract: The Askey–Wilson polynomials are a four‐parameter family of orthogonal symmetric Laurent polynomials R 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 − 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.