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Hypergeometric origins of Diophantine properties associated with the Askey scheme


Authors: Yang Chen and Mourad E. H. Ismail
Journal: Proc. Amer. Math. Soc. 138 (2010), 943-951
MSC (2000): Primary 33C20, 33C45
DOI: https://doi.org/10.1090/S0002-9939-09-10106-5
Published electronically: October 23, 2009
MathSciNet review: 2566561
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Abstract: The ``Diophantine'' properties of the zeros of certain polynomials in the Askey scheme, recently discovered by Calogero and his collaborators, are explained, with suitably chosen parameter values, in terms of the summation theorem of hypergeometric series. Here the Diophantine property refers to integer valued zeros. It turns out that the same procedure can also be applied to polynomials arising from the basic hypergeometric series. We found, with suitably chosen parameters and certain $ q$-analogues of the summation theorems, zeros of these polynomials explicitly which are no longer integer valued. This goes beyond the results obtained by the authors previously mentioned.


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Additional Information

Yang Chen
Affiliation: Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, United Kingdom
Email: ychen@ic.ac.uk

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: ismail@math.ucf.edu

DOI: https://doi.org/10.1090/S0002-9939-09-10106-5
Keywords: Generalized hypergeometric series, basic hypergeometric series, summation theorems.
Received by editor(s): March 7, 2009
Received by editor(s) in revised form: June 12, 2009
Published electronically: October 23, 2009
Communicated by: Walter Van Assche
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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