The connection between continued fractions and orthogonality which is familiar for J-fractions and T -fractions is extended to what we call R-fractions of type I and II. These continued fractions are associated with recurrence relations that correspond to multipoint rational interpolants. A Favard type theorem is proved for each type. We then study explicit models which lead to biorthogonal rational functions.Running title: Continued R-fractions 1990 Mathematics Subject Classification: Primary 40A15, 41A20 Secondary 33C45, 33D45.

Abstract. We characterize the solutions of the indeterminate moment problem associated with the continuous g-Hermite polynomials when q > 1 in terms of their Stieltjes transforms. The extremal measures are found explicitly. An analog of the Askey-Wilson integral is evaluated. It involves integrating a kernel, similar to the Askey-Wilson kernel, against any solution of the

No abstract

No abstract

Journal of Computational and Applied Mathematics 68 (1996) 115-149. doi:10.1016/0377-0427(95)00264-22016-03-04T18:47:27

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.

No abstract

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.