Several transformation formulas for 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 ${}_8{\varphi }_7$ series and Bailey's transformation formula for three ${}_8{\varphi }_7$ 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 ${}_8{\varphi }_7$ 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.