Extensions of Ramanujan's reciprocity theorem and the Andrews--Askey integral

TL;DR

This paper extends Ramanujan's reciprocity theorem from three to seven variables and generalizes the Andrews--Askey integral to a seven-parameter $q$-integral using $q$-partial differential equations.

Contribution

It introduces novel multi-variable extensions of classical $q$-series identities and integrals, broadening their applicability and understanding.

Findings

Extended Ramanujan's reciprocity theorem to seven variables

Generalized Andrews--Askey integral to a seven-parameter $q$-integral

Demonstrated the effectiveness of $q$-partial differential equations in deriving new identities

Abstract

Ramanujan's reciprocity theorem may be considered as a three-variable extension of Jacobi's triple product identity. Using the method of $q$-partial differential equations, we extend Ramanujan's reciprocity theorem to a seven-variable reciprocity formula. The Andrews--Askey integral is a $q$-integral having four parameters with base $q$. Using the same method we extend the Andrews--Askey integral formula to a $q$-integral formula which has seven parameters with base $q$.