A new product formula for $(z;q)_\infty$, with applications to asymptotics

TL;DR

This paper introduces a novel product formula for the $q$-Pochhammer symbol $(z;q)_ finite$ using gamma functions, enabling new asymptotic analysis as $q$ approaches 1.

Contribution

It provides a new infinite product representation of $(z;q)_inite$ in terms of gamma functions, extending previous work on special functions.

Findings

Derived an infinite product formula for $(z;q)_inite$

Obtained asymptotic expansions as $q$ approaches 1

Connected the formula to gamma functions and asymptotic analysis

Abstract

We express the $q$-Pochhammer symbol $(z;q)_\infty$ as an infinite product of gamma functions, analogously to how Narukawa expressed the elliptic gamma function as an infinite product of hyperbolic gamma functions. This identity is used to obtain asymptotic expansions when $q$ tends to $1$.