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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Contiguous relations, continued fractions
and orthogonality


Authors: Dharma P. Gupta and David R. Masson
Journal: Trans. Amer. Math. Soc. 350 (1998), 769-808
MSC (1991): Primary 33D45, 40A15, 39A10, 47B39
MathSciNet review: 1407490
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Abstract: We examine a special linear combination of balanced very-well-poised ${_{10} \phi _{9}}$ basic hypergeometric series that is known to satisfy a transformation. We call this $\Phi $ and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for $\Phi $ we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle's theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson's results for rational biorthogonality, Watson's $q$-analogue of Ramanujan's Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new $q$-series identities are also obtained. One is an important three-term transformation for $\Phi $'s which generalizes all the known two- and three-term $ {_{8} \phi _{7}}$ transformations. Others are new and unexpected quadratic identities for these very-well-poised ${_{8} \phi _{7}}$'s.


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Additional Information

Dharma P. Gupta
Affiliation: Department of Mathematics, University of Toronto, Toronto, M5S 3G3, Canada

David R. Masson
Affiliation: Department of Mathematics, University of Toronto, Toronto, M5S 3G3, Canada
Email: masson@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01879-0
PII: S 0002-9947(98)01879-0
Keywords: Contiguous relations, difference equations, minimal solution, continued fractions, biorthogonal rational functions, three-term-transformation, quadratic identities
Received by editor(s): November 21, 1995
Received by editor(s) in revised form: July 2, 1996
Additional Notes: Research partially supported by NSERC (Canada)
Article copyright: © Copyright 1998 American Mathematical Society