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

We express the $q$-Pochhammer symbol $(z;q{)}_{\mathrm{\infty }}$ 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 $1$.