A symmetric formula for hypergeometric series

TL;DR

This paper derives a symmetric formula for hypergeometric series using Dougall's $_2H_2$ identity and series rearrangement, connecting it to known summation formulas and demonstrating their equivalence.

Contribution

It introduces a symmetric formula for hypergeometric series and links classical identities to nonterminating summation formulas, showing their equivalence.

Findings

Derived a symmetric hypergeometric series formula.

Connected Dougall's $_2H_2$ identity to Saalsch"utz's theorem.

Linked Bailey's $_6\,\psi_6$ identity to Jackson's $_8\phi_7$ summation.

Abstract

In terms of Dougall's $_2H_2$ series identity and the series rearrangement method, we establish an interesting symmetric formula for hypergeometric series. Then it is utilized to derive a known nonterminating form of Saalsch\"{u}tz's theorem. Similarly, we also show that Bailey's $_6\psi_6$ series identity implies the nonterminating form of Jackson's $_8\phi_7$ summation formula. Considering the reversibility of the proofs, it is routine to show that Dougall's $_2H_2$ series identity is equivalent to a known nonterminating form of Saalsch\"{u}tz's theorem and Bailey's $_6\psi_6$ series identity is equivalent to the nonterminating form of Jackson's $_8\phi_7$ summation formula.