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Diophantine properties of orthogonal polynomials and rational functions


Authors: Mourad E. H. Ismail and Mizan Rahman
Journal: Proc. Amer. Math. Soc. 145 (2017), 2427-2440
MSC (2010): Primary 33C20; Secondary 33C45
DOI: https://doi.org/10.1090/proc/12355
Published electronically: February 20, 2017
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Abstract: Calogero and his collaborators recently observed that some hypergeometric polynomials can be factored as a product of two polynomials, one of which is factored into a product of linear terms. Chen and Ismail showed that this property prevails through all polynomials in the Askey scheme. We show that this factorization property is also shared by the associated Wilson and Askey-Wilson polynomials and some biorthogonal rational functions. This is applied to a specific model of an isochronous system of particles with small oscillations around the equilibrium position.


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Additional Information

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: mourad.eh.ismail@gmail.com

Mizan Rahman
Affiliation: Deceased, formerly Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada

DOI: https://doi.org/10.1090/proc/12355
Keywords: Associated Wilson polynomials, associated Askey-Wilson polynomials, biorthogonal rational functions, $R_{II}$ functions, factorization, contiguous relations, isochronous systems
Received by editor(s): June 9, 2013
Received by editor(s) in revised form: September 6, 2013
Published electronically: February 20, 2017
Additional Notes: This research was supported by Research Grants Council of Hong Kong under contract #101411
Communicated by: Walter Van Assche
Article copyright: © Copyright 2017 American Mathematical Society