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

DOI:
https://doi.org/10.1090/S0002-9947-98-01879-0

MathSciNet review:
1407490

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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