  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.
