Bailey's very well-poised -series identity

By means of Abel's method on summation by parts, some two term recurrence relations on very well-poised -series are established. Their iteration yields a -series transformation with an extra natural number parameter. Evaluating the limiting series via Jacobi's triple product identity, we are led surprisingly to the celebrated bilateral -series identity discovered by Bailey (1936). Then we shall further generalize it to a very well-poised -series identity, which contains Shukla's formula (1959) as special case. Finally, the Abel's method on summation by parts will be employed again to investigate the bibasic hypergeometric series summation, which may be considered as an extension of a “split-poised” transformation on terminating -series due to Gasper (1989).