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Non-Terminating Basic Hypergeometric Series and the q-Zeilberger Algorithm

Published online by Cambridge University Press:  12 December 2008

William Y. C. Chen ,
Qing-Hu Hou  and
Yan-Ping Mu
William Y. C. Chen
Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People's Republic of China (chen@nankai.edu.cn; hou@nankai.edu.cn)
Qing-Hu Hou
Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People's Republic of China (chen@nankai.edu.cn; hou@nankai.edu.cn)
Yan-Ping Mu
Affiliation:
College of Science, Tianjin University of Technology, Tianjin 300384, People's Republic of China (yanping.mu@gmail.com)
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Abstract

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We present a systematic method for proving non-terminating basic hypergeometric identities. Assume that k is the summation index. By setting a parameter x to xqn, we may find a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all non-terminating basic hypergeometric summation formulae in the work of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulae, including the Sears–Carlitz transformation, transformations of the very well-poised 8φ7 series, the Rogers–Fine identity and the limiting case of Watson's formula that implies the Rogers–Ramanujan identities.

Keywords

basic hypergeometric seriesq-Zeilberger algorithmBailey's very well-poised 6ψ6 summation formulaSears–Carlitz transformationRogers–Fine identity
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 51 , Issue 3 , October 2008 , pp. 609 - 633
Copyright
Copyright © Edinburgh Mathematical Society 2008