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Hypergeometric origins of Diophantine properties associated with the Askey scheme
Author(s):
Yang
Chen;
Mourad
E. H.
Ismail
Journal:
Proc. Amer. Math. Soc.
MSC (2000):
Primary 33C20, 33C45
Posted:
October 23, 2009
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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  -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:
10.1090/S0002-9939-09-10106-5
PII:
S 0002-9939(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
Posted:
October 23, 2009
Communicated by:
Walter Van Assche
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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