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Nonterminating q-Whipple Transformations for Basic Hypergeometric Series in U(n)

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Book cover Partitions, q-Series, and Modular Forms

Part of the book series: Developments in Mathematics ((DEVM,volume 23))

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Abstract

In this paper we derive multivariable generalizations of Bailey’s classical nonterminating q-Whipple and q-Saalschütz transformations. We work in the setting of multiple basic hypergeometric series very-well-poised on unitary groups U(n + 1), multiple series that are associated to the root system A n . We extend Bailey’s proofs of these transformations by first taking suitable limits of our U(n+ 1) 10ϕ9 transformation formula, in which the multiple sums are taken over an n-dimensional tetrahedron (n-simplex). A natural partition of the (finite) n-simplex combines with our analysis of the convergence of the multiple series to yield our transformations. We expect that all of these results will directly extend to the analogous case of multiple basic hypergeometric series associated to the root system D n .

MathematicsSubject Classification: Primary: 33D70, 05A19; Secondary: 05A30

The first author was partially supported by NSF grants DMS 86-04232, DMS 89-04455, DMS 90-96254, NSA supplements to these NSF grants, by NSF grant DMS 0100288, and by NSA grants MDA 904-88-H-2010, MDA 904-91-H-0055, MDA 904-93-H-3032, MDA 904-97-1-0019, MDA 904-99-1-0003, H98230-06-1-0064, and H98230-08-1-0093. Both authors were partially supported by NSA grant MDA 904-91-H-0055.

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  1. Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, OH, 43210, USA

    Stephen C. Milne & John W. Newcomb

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  1. Stephen C. Milne
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Correspondence to Stephen C. Milne .

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  1. , Department of Mathematics, University of Florida, Gainesville, 32611, Florida, USA

    Krishnaswami Alladi

  2. , Mathematics, University of Florida, Gainesville, 32611, Florida, USA

    Frank Garvan

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Milne, S.C., Newcomb, J.W. (2012). Nonterminating q-Whipple Transformations for Basic Hypergeometric Series in U(n). In: Alladi, K., Garvan, F. (eds) Partitions, q-Series, and Modular Forms. Developments in Mathematics, vol 23. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0028-8_12

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