On ${}_5\ddot{I}{ˆ}_5$ identities of Bailey

Abstract

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

10 pages