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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Contiguous relations, continued fractions and orthogonality
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by Dharma P. Gupta and David R. Masson PDF
Trans. Amer. Math. Soc. 350 (1998), 769-808 Request permission

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
  • 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)
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 769-808
  • MSC (1991): Primary 33D45, 40A15, 39A10, 47B39
  • DOI: https://doi.org/10.1090/S0002-9947-98-01879-0
  • MathSciNet review: 1407490