A new recurrence relation for the truncated very-well-poised $_6\psi_6$ series and Bailey's summation formula

TL;DR

The paper introduces a truncated version of the very-well-poised $_6 heta_6$ series, establishes a recurrence relation for it, and provides an elementary proof of Bailey's $_6 heta_6$ summation formula.

Contribution

It presents a new recurrence relation for the truncated $_6 heta_6$ series and offers an elementary proof of Bailey's summation formula.

Findings

Established a recurrence relation for the truncated $_6 heta_6$ series.

Provided an elementary proof of Bailey's $_6 heta_6$ summation formula.

Enhanced understanding of the structure of very-well-poised basic hypergeometric series.

Abstract

In this paper we introduce the so-called truncated very-well-poised $_6\psi_6$ series and set up an explicit recurrence relation for it by means of the classical Abel lemma on summation by parts. This new recurrence relation implies an elementary proof of Bailey's well-known $_6\psi_6$ summation formula.