# The united proofs for three $q$-extensions of Dougall's $_2H_2$ summation formula

## TL;DR

This paper provides unified proofs for three $q$-extensions of Dougall's $_2H_2$ summation formula using analytic continuation, and discusses related results in the context of basic hypergeometric series.

## Contribution

It introduces a unified proof approach for multiple $q$-extensions of Dougall's $_2H_2$ summation formula, expanding understanding of basic hypergeometric identities.

## Key findings

- Unified proofs for three $q$-extensions of Dougall's $_2H_2$ formula
- Discussion of related hypergeometric results
- Application of analytic continuation method

## Abstract

In terms of the analytic continuation method, we give the united proofs for three $q$-extensions of Dougall's $_2H_2$ summation formula. Some related results are also discussed in this paper.