Generalizations of a terminating summation formula of basic hypergeometric series and their applications

TL;DR

This paper extends a known summation formula for basic hypergeometric series to more general forms, including bilateral sums, and demonstrates their applications by unifying classical identities.

Contribution

It introduces new unilateral and bilateral summation formulas for basic hypergeometric series, linking them to classical identities and expanding their applicability.

Findings

Derived a $q$-analogue of a $_4F_3$-summation formula.

Unified Jacobi's triple product and quintuple product identities.

Established new bilateral summation identities.

Abstract

We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula. And, an identity unifying Jacobi's triple product identity and the quintuple product identity is obtained as a special case of the bilateral one.