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A simple proof of Bailey's very-well-poised ${}_{6}\psi_{6}$ summation


Author: Michael Schlosser
Journal: Proc. Amer. Math. Soc. 130 (2002), 1113-1123
MSC (2000): Primary 33D15
DOI: https://doi.org/10.1090/S0002-9939-01-06175-5
Published electronically: October 1, 2001
MathSciNet review: 1873786
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Abstract: We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauß' $_2F_1$ summation and elementary series manipulations to give a simple proof of Dougall's $_2H_2$ summation. Similarly, we apply Rogers' nonterminating $_6\phi_5$ summation and elementary series manipulations to give a simple proof of Bailey's very-well-poised $_6\psi_6$ summation. Our method of proof extends M. Jackson's first elementary proof of Ramanujan's $_1\psi_1$ summation.


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Additional Information

Michael Schlosser
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Address at time of publication: Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
Email: mschloss@math.ohio-state.edu, schlosse@ap.univie.ac.at

DOI: https://doi.org/10.1090/S0002-9939-01-06175-5
Keywords: Bilateral basic hypergeometric series, $q$-series, Ramanujan's $_1\psi_1$ summation, Dougall's $_2H_2$ summation, Bailey's $_6\psi_6$ summation
Received by editor(s): July 7, 2000
Received by editor(s) in revised form: September 25, 2000, and October 18, 2000
Published electronically: October 1, 2001
Communicated by: John R. Stembridge
Article copyright: © Copyright 2001 American Mathematical Society

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