Contiguous relations, continued fractions and orthogonality
Authors:
Dharma P. Gupta and David R. Masson
Journal:
Trans. Amer. Math. Soc. 350 (1998), 769808
MSC (1991):
Primary 33D45, 40A15, 39A10, 47B39
MathSciNet review:
1407490
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Abstract: We examine a special linear combination of balanced verywellpoised basic hypergeometric series that is known to satisfy a transformation. We call this and show that it satisfies certain threeterm contiguous relations. From two of these contiguous relations for we obtain fiftysix pairwise linearly independent solutions to a threeterm recurrence that generalizes the recurrence for AskeyWilson 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 threeterm transformation for 's which generalizes all the known two and threeterm transformations. Others are new and unexpected quadratic identities for these verywellpoised '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
DOI:
http://dx.doi.org/10.1090/S0002994798018790
PII:
S 00029947(98)018790
Keywords:
Contiguous relations,
difference equations,
minimal solution,
continued fractions,
biorthogonal rational functions,
threetermtransformation,
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
