Several transformation formulas for basic hypergeometric series

TL;DR

This paper extends classical hypergeometric series identities to six-variable cases, providing new transformation formulas and generalizations of Ramanujan's reciprocity theorem, enriching the theory of basic hypergeometric series.

Contribution

It introduces a six-variable generalization of Andrews' identity and Ramanujan's reciprocity theorem, expanding the scope of hypergeometric series transformations.

Findings

Established a six-variable generalization of Andrews' identity.

Derived a six-variable generalization of Ramanujan's reciprocity theorem.

Presented new results involving bilateral basic hypergeometric series.

Abstract

In 1981, Andrews gave a four-variable generalization of Ramanujan's ${_1\psi_1}$ summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two ${_8\phi_7}$ series and Bailey's transformation formula for three ${_8\phi_7}$ series. Then it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two $_3\psi_3$ summation formulas in terms of two limiting relations and Bailey's another transformation formula for three $_8\phi_7$ series. Based on the two limiting relations, some different results involving bilateral basic hypergeometric series are also deduced from the Guo--Schlosser transformation formula and other two transformation formulas.