The Bailey Transform And Some New Q-series Identities

In [11], W. N. Bailey first presented Bailey's transform [27]:IfThen,Subject to conditions on the four sequencesαn,βn,γn andδn, which make all the infinite series absolutely convergent. By using Bailey'transform, we can prove the following five identities:As we known, the first two identities are new. Furthermore, combined q-binomial theorem with generating function, we gave the following new identity:The above identity is a generalization in Ramanujan's "Lost Notebook" [20, p.13] (z=1).