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Askey-Wilson type functions with bound states

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Haine, Luc [UCL] Iliev, P

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of q-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Grunbaum.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 2006
Language Anglais
Journal information "The Ramanujan Journal : an international journal devoted to areas of mathematics influenced by Ramanujan" - Vol. 11, no. 3, p. 285-329 (2006)
Peer reviewed yes
Publisher Springer (Dordrecht)
issn 1382-4090
e-issn 1572-9303
Publication status Publié
Affiliation UCL - SC/MATH - Département de mathématique
Keywords basic orthogonal polynomials and functions ; Askey-Wilson polynomials
Links
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Bibliographic reference Haine, Luc ; Iliev, P. Askey-Wilson type functions with bound states. In: The Ramanujan Journal : an international journal devoted to areas of mathematics influenced by Ramanujan, Vol. 11, no. 3, p. 285-329 (2006)
Permanent URL http://hdl.handle.net/2078.1/38375