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On ${}_5ψ_5$ identities of Bailey

Aritram Dhar

TL;DR

The paper analyzes Bailey's bilateral summations ${}_5\psi_5$ by deriving two identities from Carlitz's terminating ${}_5\phi_4$ through even/odd parameter substitutions, then shows that in the limit $n\to\infty$ these yield two ${}_3\psi_3$ summations with explicit infinite-product forms. It further provides Ismail-type proofs of these ${}_3\psi_3$ identities by recasting them as finite ${}_3\phi_2$ sums and employing analytic continuation arguments inspired by Ramanujan's ${}_1\psi_1$ and Bailey's ${}_6\psi_6$ framework. The work highlights a deep connection between bilateral and unilateral basic hypergeometric series and demonstrates how Carlitz-type and Ismail-style techniques complement each other to establish exact summations. The results extend Bailey's identities, offering multiple corroborating pathways (finite ${}_5\phi_4$ transforms and infinite ${}_3\psi_3$ products) with potential applications in related areas of basic hypergeometric analysis and $q$-series theory.

Abstract

In this paper, we provide proofs of two ${}_5ψ_5$ summation formulas of Bailey using a ${}_5φ_4$ identity of Carlitz. We show that in the limiting case, the two ${}_5ψ_5$ identities give rise to two ${}_3ψ_3$ summation formulas of Bailey. Finally, we prove the two ${}_3ψ_3$ identities using a technique initially used by Ismail to prove Ramanujan's ${}_1ψ_1$ summation formula and later by Ismail and Askey to prove Bailey's very-well-poised ${}_6ψ_6$ sum.

On ${}_5ψ_5$ identities of Bailey

TL;DR

The paper analyzes Bailey's bilateral summations by deriving two identities from Carlitz's terminating through even/odd parameter substitutions, then shows that in the limit these yield two summations with explicit infinite-product forms. It further provides Ismail-type proofs of these identities by recasting them as finite sums and employing analytic continuation arguments inspired by Ramanujan's and Bailey's framework. The work highlights a deep connection between bilateral and unilateral basic hypergeometric series and demonstrates how Carlitz-type and Ismail-style techniques complement each other to establish exact summations. The results extend Bailey's identities, offering multiple corroborating pathways (finite transforms and infinite products) with potential applications in related areas of basic hypergeometric analysis and -series theory.

Abstract

In this paper, we provide proofs of two summation formulas of Bailey using a identity of Carlitz. We show that in the limiting case, the two identities give rise to two summation formulas of Bailey. Finally, we prove the two identities using a technique initially used by Ismail to prove Ramanujan's summation formula and later by Ismail and Askey to prove Bailey's very-well-poised sum.
Paper Structure (8 sections, 7 theorems, 41 equations)

This paper contains 8 sections, 7 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Proofs of the Two ${}_5\psi_5$ Identities
  3. Proof of Theorem \ref{['5psi51']}
  4. Proof of Theorem \ref{['5psi52']}
  5. Two Limiting Cases
  6. Ismail Type Proofs of the Two ${}_3\psi_3$ Identities
  7. Proof of Theorem \ref{['3psi31']}
  8. Proof of Theorem \ref{['3psi32']}

Key Result

Theorem 1.1

(Bailey 2) For any non-negative integer $n$, where $bcde=q^{n+1}$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7