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In this paper, we provide proofs of two ${}_5\psi_5$ summation formulas ofBailey using a ${}_5\phi_4$ identity of Carlitz. We show that in the limitingcase, the two ${}_5\psi_5$ identities give rise to two ${}_3\psi_3$ summationformulas of Bailey. Finally, we prove the two ${}_3\psi_3$ identities using atechnique initially used by Ismail to prove Ramanujan's ${}_1\psi_1$ summationformula and later by Ismail and Askey to prove Bailey's very-well-poised${}_6\psi_6$ sum.

On ${}_5\psi_5$ identities of Bailey