## 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.## References

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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