Narrow Results

In this paper, we provide proofs of two ${}_5{\psi }_5$ summation formulas of Bailey using a ${}_5{\varphi }_4$ identity of Carlitz. We show that in the limiting case, the two ${}_5{\psi }_5$ identities give rise to two ${}_3{\psi }_3$ summation formulas of Bailey. Finally, we prove the two ${}_3{\psi }_3$ identities using a technique initially used by Ismail to prove Ramanujan’s ${}_1{\psi }_1$ summation formula and later by Ismail and Askey to prove Bailey’s very-well-poised ${}_6{\psi }_6$ sum.