A simple proof of Bailey’s very-well-poised ₆𝜓₆ summation

Schlosser, Michael

doi:10.1090/S0002-9939-01-06175-5

A simple proof of Bailey’s very-well-poised ${}_{6}\psi _{6}$ summation
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Proc. Amer. Math. Soc. 130 (2002), 1113-1123 Request permission

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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