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Contiguous relations, continued fractions and orthogonality |
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| Authors: | Dharma P. Gupta David R. Masson |
| Affiliation: | Department of Mathematics, University of Toronto, Toronto, M5S 3G3, Canada David R. Masson ; Department of Mathematics, University of Toronto, Toronto, M5S 3G3, Canada |
| Abstract: | We examine a special linear combination of balanced very-well-poised  basic hypergeometric series that is known to satisfy a transformation. We call this  and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for  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  -analogue of Ramanujan's Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new  -series identities are also obtained. One is an important three-term transformation for  's which generalizes all the known two- and three-term  transformations. Others are new and unexpected quadratic identities for these very-well-poised  's. |
| Keywords: | Contiguous relations difference equations minimal solution continued fractions biorthogonal rational functions three-term-transformation quadratic identities |
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