A common recursive formula for Ramanujan's {}_1\psi_1 and Bailey's very-well-poised {}_6\psi_6 summation formulas.

In this paper, we establish a general recursive formula via the usual telescoping method for certain bilateral truncated sum which may serve as a common source for Ramanujan's {}_1\psi _1, Bailey's very-well-poised {}_6\psi _6, and Rogers' {}_6\phi _5 summation formulas. The corresponding derivations for these summation formulas are presented in details. [ABSTRACT FROM AUTHOR]