A First Order $q$-Difference System for the $BC_1$-Type Jackson Integral and Its Applications

TL;DR

This paper derives a first order $q$-difference system for the $BC_1$-type Jackson integral, providing explicit equations and applications to hypergeometric summation and product formulas, advancing the understanding of $q$-series and integrals.

Contribution

It introduces a first order $q$-difference system for the $BC_1$-type Jackson integral with explicit basis, enabling simplified proofs of known hypergeometric identities.

Findings

Explicit $q$-difference system for $BC_1$-type Jackson integral

Simplified proof of Gustafson's hypergeometric summation formula

Derivation of the Nassrallah-Rahman and Gustafson product formula

Abstract

We present an explicit expression for the $q$-difference system, which the $BC_1$-type Jackson integral ($q$-series) satisfies, as first order simultaneous $q$-difference equations with a concrete basis. As an application, we give a simple proof for the hypergeometric summation formula introduced by Gustafson and the product formula of the $q$-integral introduced by Nassrallah-Rahman and Gustafson.