A generalization of the q-(Pfaff)-Saalschutz summation formula is proved. This
implies a generalization of the Burge transform, resulting in an additional dimension of
the ``Burge tree''. Limiting cases of our summation formula imply the (higher-level) Bailey
lemma, provide a new decomposition of the q-multinomial coefficients, and can be used to
prove the Lepowsky and Primc formula for the A_1^{(1)} string functions.