# On ${}_5\psi_5$ identities of Bailey

## TL;DR

This paper proves new ${}_5 ext{psi}_5$ identities of Bailey using a ${}_5 ext{phi}_4$ identity, and derives related ${}_3 ext{psi}_3$ formulas through limiting cases, employing techniques from Ismail and Askey.

## Contribution

It introduces novel proofs of Bailey's ${}_5 ext{psi}_5$ identities and connects them to ${}_3 ext{psi}_3$ formulas using established summation techniques.

## Key findings

- Proved two ${}_5 ext{psi}_5$ summation formulas of Bailey.
- Derived two ${}_3 ext{psi}_3$ summation formulas as limits.
- Applied techniques from Ismail and Askey to establish identities.

## Abstract

In this paper, we provide proofs of two ${}_5\psi_5$ summation formulas of Bailey using a ${}_5\phi_4$ 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.