  The Ramanujan $_1\psi_1$ summation theorem in studied from the perspective of
$q$-Jackson integrals, $q$-difference equations and connection formulas. This
is an approach which has previously been shown to yield Bailey's
very-well-poised $_6\psi_6$ summation. Bilateral Jackson integral
generalizations of the Dixon--Anderson and Selberg integrals relating to the
type $A$ root system are identified as natural candidates for multidimensional
generalizations of the Ramanujan $_1\psi_1$ summation theorem. New results of
this type are announced, and furthermore they are put into context by reviewing
from previous literature explicit product formulas for Jackson integrals
relating to other roots systems obtained from the same perspective.
