Abstract. We examine a special linear combination of balanced very-wellpoised 10 φ 9 basic hypergeometric series that is known to satisfy a transformation. We call this Φ and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for Φ 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 Φ's which generalizes all the known two-and three-term 8 φ 7 transformations. Others are new and unexpected quadratic identities for these very-well-poised 8 φ 7 's.
Abstract. We generalize Watson's q-analogue of Ramanujan's Entry 40 continued fraction by deriving solutions to a 10 φ 9 series contiguous relation and applying Pincherle's theorem. Watson's result is recovered as a special terminating case, while a limit case yields a new continued fraction associated with an 8 φ 7 series contiguous relation.
A 10 φ 9 contiguous relation is used to derive contiguous relations for a very-well-poised 8 φ 7 . These in turn yield solutions to the associated q -Askey-Wilson polynomial recurrence relation, expressions for the associated continued fraction, the weight function and a q -analogue of a generalized Dougall's theorem.
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