On a theta product of Jacobi and its applications to $q$-gamma products

TL;DR

This paper presents a new proof of a Jacobi product formula, connects it to a $q$-trigonometric product, and derives a $q$-analogue of the Gauss multiplication formula, with explicit formulas for $q$-gamma products.

Contribution

It provides a novel proof of a Jacobi product formula and applies it to develop new $q$-analogues of classical gamma function identities.

Findings

Established a new proof for a Jacobi product formula

Derived a $q$-analogue of the Gauss multiplication formula

Provided explicit formulas for short products of $q$-gamma functions

Abstract

We give a new proof for a product formula of Jacobi which turns out to be equivalent to a $q$-trigonometric product which was stated without proof by Gosper. We apply this formula to derive a $q$-analogue for the Gauss multiplication formula for the gamma function. Furthermore, we give explicit formulas for short products of $q$-gamma functions.