A New Product Formula for $$(z;q)_\infty $$, with Applications to Asymptotics

Abstract We express the q -Pochhammer symbol $$(z;q)_\\infty $$ ( z ; q ) ∞ 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.

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Publication Details

Journal
Constructive Approximation
Published
2026-09-19
DOI
https://doi.org/10.1007/s00365-026-09783-2
Primary Topic
Advanced Mathematical Identities
Type
article
Field-Weighted Citation Impact
0.00
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article

A New Product Formula for $$(z;q)_\infty $$, with Applications to Asymptotics

Hjalmar Rosengren, Arash Arabi Ardehali
Constructive Approximation
Advanced Mathematical Identities
article

A New Product Formula for $$(z;q)_\infty $$, with Applications to Asymptotics

Hjalmar Rosengren, Arash Arabi Ardehali
article en

Abstract

Abstract We express the q -Pochhammer symbol $$(z;q)_\infty $$ ( z ; q ) ∞ 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.

Constructive Approximation
Sharif University of Technology (IR), Chalmers University of Technology (SE), University of Gothenburg (SE)
Openalex Percentile: Top 77%
Advanced Mathematical Identities
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