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

Arash Arabi Ardehali; Hjalmar Rosengren

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

Quantum Algebra 2026-02-27 v2 High Energy Physics - Theory Mathematical Physics Classical Analysis and ODEs math.MP

Authors: Arash Arabi Ardehali , Hjalmar Rosengren

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 $1$ .

Keywords

q-Pochhammer symbol gamma function elliptic gamma function asymptotic expansion hyperbolic gamma function

Cite

@article{arxiv.2602.11329,
  title  = {A new product formula for $(z;q)_\infty$, with applications to asymptotics},
  author = {Arash Arabi Ardehali and Hjalmar Rosengren},
  journal= {arXiv preprint arXiv:2602.11329},
  year   = {2026}
}

Comments

19 pages, 3 figures; v2: new appendix on connection with Narukawa's identity

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

Abstract

Keywords

Cite

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