A simple proof of Bailey's very-well-poised ~6psi~6 summation

We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauss' 2F1 summation and elementary series manipulations to give a simple proof of Dougall's 2 H 2 summation. Similarly, we apply Rogers' nonterminating 6o5 summation and elementary series manipulations to give a simple proof of Bailey's very-well-poised 6ψ6 summation. Our method of proof extends M. Jackson's first elementary proof of Ramanujan's 1ψ1 summation.