Generalizations of Ramanujan's reciprocity formula and the Askey-Wilson integral

TL;DR

This paper extends classical identities in basic hypergeometric series, deriving multi-variable generalizations of Ramanujan's reciprocity formula and the Askey-Wilson integral using transformation formulas and known identities.

Contribution

It introduces new multi-variable generalizations of key hypergeometric identities, expanding their applicability and theoretical understanding.

Findings

Extended Bailey's $_6\psi_6$-series identity to multi-variable form

Derived multi-variable Ramanujan reciprocity formula

Generalized the Askey-Wilson integral to multiple variables

Abstract

By using two known transformation formulas for basic hypergeometric series, we establish a direct extension of Bailey's $_6\psi_6$-series identity. Subsequently, it and Milne's identity are employed to drive multi-variable generalizations of Ramanujan's reciprocity formula. Then we utilize also Milne's identity to deduce a multi-variable generalization of the Askey-Wilson integral.