On integral representations of q-gamma and q-beta functions

TL;DR

This paper explores q-integral representations of q-gamma and q-beta functions, revealing a new q-constant and providing simplified proofs for classical identities like Jacobi's and Ramanujan's formulas.

Contribution

It introduces novel q-integral representations and a new q-constant, offering simplified proofs of key identities in q-series and hypergeometric functions.

Findings

Introduction of a new q-constant

Simplified proofs of Jacobi's and Ramanujan's identities

Enhanced understanding of q-gamma and q-beta functions

Abstract

We study q-integral representations of the q-gamma and the q-beta functions. This study leads to a very interesting q-constant. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series.