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.

Keywords:

• Basic hypergeometric series
• Ramanujan's ${}_1{\psi }_1$ summation formula
• Andrews' identity
• Ramanujan's reciprocity theorem

2010 Mathematics Subject Classifications:

• Primary: 05A19
• Secondary: 33D15

Acknowledgments

The author is grateful to the reviewer for helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is supported by the National Natural Science Foundation of China [grant numbers 12071103, 11661032, 11601151].