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

Arash Arabi Ardehali, Hjalmar Rosengren

TL;DR

This paper derives a gamma-product representation of the $q$-Pochhammer symbol $(z;q)_\infty$, focusing on the non-modular case $(e^{-y};e^{-eta})_\infty$ and its $q\to1$ asymptotics. It presents two proofs—the Poisson-summation method and an elementary Artin-identity approach—both yielding the same infinite product over $\Gamma$-functions dressed by polylogarithmic factors, with explicit $\operatorname{Li}_2$ and $\coth$-type terms. The authors develop a uniform asymptotic theory for $\log (e^{-y};e^{-eta})_\infty$ as $\beta\to0$, covering scaling regimes $y\sim x\beta^c$ for $c\ge0$, including new cases $0<c<1$ and $c>1$, and provide detailed error estimates. They connect these expansions to modular transformations in the special case $y=\beta$ (Dedekind's eta) and to quantum-field-theoretic BPS partition functions, illustrating the broader physical relevance.Overall, the work provides robust, uniform $q\to1$ asymptotics for q-Pochhammer symbols across multiple scaling regimes and clarifies the analytic structure via polylogarithms and gamma-function products.

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 $1$.

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

TL;DR

This paper derives a gamma-product representation of the -Pochhammer symbol , focusing on the non-modular case and its asymptotics. It presents two proofs—the Poisson-summation method and an elementary Artin-identity approach—both yielding the same infinite product over -functions dressed by polylogarithmic factors, with explicit and -type terms. The authors develop a uniform asymptotic theory for as , covering scaling regimes for , including new cases and , and provide detailed error estimates. They connect these expansions to modular transformations in the special case (Dedekind's eta) and to quantum-field-theoretic BPS partition functions, illustrating the broader physical relevance.Overall, the work provides robust, uniform asymptotics for q-Pochhammer symbols across multiple scaling regimes and clarifies the analytic structure via polylogarithms and gamma-function products.

Abstract

We express the -Pochhammer symbol 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 tends to .
Paper Structure (5 sections, 3 theorems, 98 equations, 3 figures)

This paper contains 5 sections, 3 theorems, 98 equations, 3 figures.

Table of Contents

  1. Introduction
  2. First proof of the main identity
  3. Alternative proof of the main identity
  4. Applications
  5. Error analysis

Key Result

Corollary 1

For $N\in\mathbb Z_{>0}$, $\operatorname{Re} \beta >0$ and $y\notin 2\pi \mathrm{i}\mathbb Z$, where $f_N$ is defined by stirling-general.

Figures (3)

  • Figure 1: Error of the partial sums of \ref{['uniform-asymptotics']} as a function of the truncation order. We take $y=x\beta^c$, with $x=3$, $\beta=1/16$ and the four values $c=0,\,1/2,\, 1,\, 2$. The dashed lines are the estimates for the optimal truncation order $N_\ast$ and truncation error $R_\ast$ computed from \ref{['eq:uniform-kast-Tkast']}.
  • Figure 2: Error of the partial sums of \ref{['asymptotics-four-regimes']} as a function of the truncation order. We take $y=x\beta^c$, with $x=3$, $\beta=1/16$ and $c=0,\,1/2,\, 1,\, 2$. For the cases $c\neq 1$, the dashed lines are the estimates for the optimal truncation order $N_\ast$ and truncation error $R_\ast$ computed from \ref{['eq:kast-fixed-y']}, \ref{['eq:kast-small-c']} and \ref{['eq:kast-c>1']}. For $c=1$, the dotted line indicates the exact remainder term \ref{['exact-remainder']}.
  • Figure 3: Truncation errors in the case $y=x\beta$, with $\beta=1/16$ and $x=2.9$. To the left, the error in the uniform expansion \ref{['uniform-asymptotics']}, with estimates for the optional truncation order and error from \ref{['eq:uniform-kast-Tkast']}. To the right, the error in the complete asymptotic expansion \ref{['asymptotics-c-1']}, with estimates from \ref{['eq:kast-c=1']}.

Theorems & Definitions (5)

  • Corollary 1
  • Corollary 2
  • proof
  • Corollary 3
  • proof