Curious extensions of Ramanujan's 1-psi-1 summation formula

We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our identities, by analytic continuation, to bilateral summation formulae which contain Ramanujan's 1-psi-1 summation and a very-well-poised 4-psi-6 summation as special cases.