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

This paper derives an infinite product formula for the $q$-Pochhammer symbol in terms of gamma functions, analogous to Narukawa's expression for the elliptic gamma function, and utilizes this identity to establish asymptotic expansions as $q$ approaches 1.

The Gist

Imagine you are trying to understand a very complex, mysterious machine. In the world of mathematics, this machine is called the q-Pochhammer symbol. It's a special kind of infinite product (a never-ending multiplication) that shows up everywhere in physics, from quantum mechanics to string theory.

For a long time, mathematicians knew how this machine behaved when you turned a specific "dial" (let's call it $q$) all the way to 1. When you do that, the machine transforms into familiar, simpler things like the exponential function ($e^x$) or the Gamma function (which is a fancy version of factorials).

However, the authors of this paper, Arash Arabi Ardahali and Hjalmar Rosengren, wanted to know: What happens in the messy middle? What does the machine look like just before it turns into those simple things?

Here is the story of their discovery, explained without the heavy math jargon.

1. The Problem: A Giant, Confusing Recipe

Think of the q-Pochhammer symbol as a recipe that calls for an infinite number of ingredients. If you try to taste the final dish (calculate the result) when the "temperature" ($q$) is close to 1, the recipe becomes incredibly hard to follow. The numbers get huge, and the calculation becomes unstable.

Mathematicians have a "Golden Rule" for factorials called Stirling's Approximation. It's like a cheat sheet that tells you how big a factorial is without actually multiplying all the numbers. The authors realized that their infinite product recipe was secretly hiding a similar cheat sheet, but it was buried deep inside.

2. The Breakthrough: The "Gamma" Mirror

The authors discovered a new way to write this infinite product. Instead of seeing it as a long list of multiplications, they found a way to express it as an infinite product of Gamma functions.

The Analogy:
Imagine you have a giant, tangled ball of yarn (the q-Pochhammer symbol). For years, people tried to untangle it by pulling on the ends, but it just got knottier.
The authors found a magic mirror. When they looked at the yarn in this mirror, the tangled ball didn't look like yarn anymore; it looked like a perfect, repeating pattern of Gamma functions (which are the "smooth, well-behaved" cousins of factorials).

They didn't just find one mirror; they built two different ways to prove the mirror exists:

• Proof 1 (The Fourier Approach): They used a technique called "Poisson Summation." Imagine you have a song playing, and you want to know its frequency. Instead of listening to the song, you look at its shadow on a wall. The shadow reveals the hidden frequencies. They did this with the math, turning the problem into a frequency puzzle and solving it.
• Proof 2 (The Elementary Approach): This was a more direct, step-by-step construction. They built the solution brick by brick using known identities, showing that if you stack these specific mathematical bricks correctly, they form the exact shape of the q-Pochhammer symbol.

3. Why Does This Matter? (The "Asymptotic" Magic)

The real power of this discovery is what happens when you turn the dial ($q$) toward 1. This is called the asymptotic limit.

In physics, this is like zooming in on a digital image. When you zoom in too far, pixels become visible, and the picture gets blocky. The authors' formula tells you exactly how the picture gets "blocky" and how to fix it.

They applied their formula to different scenarios:

• The "Slow" Zoom: When the variable changes slowly, the formula gives a smooth, predictable path.
• The "Fast" Zoom: When the variable changes rapidly, the formula reveals a chaotic-looking pattern that actually follows a very strict, hidden order.

They found that depending on how you scale the variables, the formula reveals different "regimes" (like different weather patterns). Some of these regimes were known, but they discovered new ones (specifically when the scaling factor is between 0 and 1) that were previously a mystery.

4. The Physics Connection: From 4D to 3D to 2D

The paper has a fascinating connection to the real world of physics (Quantum Field Theory).

• The Big Picture: In physics, there are objects called "multiplets" that exist in 4 dimensions.
• The Reduction: If you shrink one of those dimensions (like rolling up a long tube), the 4D object becomes a 3D object. If you shrink another, it becomes 2D.
• The Connection: The authors' formula is the mathematical translation of this physical shrinking process.
  • A famous mathematician named Narukawa had already found a formula for shrinking a 4D object to a 3D one.
  • These authors found the formula for shrinking a 3D object down to a 2D one.

It's like finding the missing link in a chain. If Narukawa's formula is the bridge from a skyscraper to a house, this new formula is the bridge from the house to a garden shed. It explains how the "energy" (or partition function) of the system changes as the dimensions collapse.

5. The "Error" Analysis: How Close is Close Enough?

Finally, the authors asked: "If we stop our infinite recipe halfway, how wrong are we?"
Since you can't multiply an infinite number of things, you have to stop at some point. This leaves an "error."
They used computer simulations to see how this error behaves. They found that:

• If you stop too early, the error is huge.
• If you stop too late, the error actually gets worse again (a common problem with infinite series).
• There is a "Sweet Spot" (an optimal stopping point) where the error is tiny.

They mapped out exactly where this sweet spot is for different scenarios, giving physicists a precise guide on how many terms they need to calculate to get a perfect answer without wasting computer power.

Summary

In simple terms, this paper is about decoding a complex mathematical recipe.

1. The Discovery: They found a new way to write a complex infinite product using simpler, well-known functions (Gamma functions).
2. The Method: They proved it using two different mathematical "languages" (one based on waves/frequencies, one based on building blocks).
3. The Application: They used this to predict exactly how the formula behaves when pushed to its limits (when $q \to 1$), revealing new patterns that help physicists understand how the universe behaves when dimensions shrink.
4. The Guide: They figured out exactly how to use this formula in practice to get the most accurate results with the least amount of effort.

It's a beautiful example of taking a messy, infinite problem and finding a clean, structured pattern hidden inside it.

Technical Summary

1. Problem Statement

The paper addresses the asymptotic behavior of the infinite $q$-Pochhammer symbol, defined as:
$$(z; q)_\infty = \prod_{j=0}^\infty (1 - zq^j)$$
as the parameter $q$ approaches 1 (specifically $q = e^{-\beta}$ with $\beta \to 0$). While the limits of this symbol are known to relate to the exponential, Gamma, and dilogarithm functions, obtaining uniform asymptotic expansions for $(z; q)_\infty$ when the variable $z$ scales with $q$ (i.e., $z = e^{-x\beta^c}$) is a non-trivial problem.

Existing literature covers specific regimes (e.g., $c=0$ or $c=1$), but a unified, rigorous product formula that facilitates the analysis of intermediate regimes ($0 < c < 1$) and large regimes ($c > 1$) was lacking. This is particularly relevant for the study of basic hypergeometric integrals and their physical interpretations in Quantum Field Theory (QFT).

2. Methodology

The authors employ a multi-faceted approach combining complex analysis, special function identities, and heuristic error analysis:

• Derivation of a New Product Identity: The core of the work is the derivation of an identity expressing $(e^{-y}; e^{-\beta})_\infty$ as an infinite product of Gamma functions. This is analogous to Narukawa's identity, which expresses the elliptic gamma function as a product of hyperbolic gamma functions.
• Two Distinct Proofs:
  1. Fourier Analysis (Poisson Summation): The first proof utilizes Binet's integral representation for the remainder term in Stirling's approximation. By applying Poisson's summation formula to a specific test function involving the remainder, they derive the product formula.
  2. Elementary Series Manipulation (Artin's Identity): The second proof avoids Fourier analysis. It relies on Artin's identity for the remainder term, expressing it as a series. By interchanging summations and utilizing integral representations of the terms, they reconstruct the product formula.
• Asymptotic Expansion: Using the new product formula, the authors derive a uniform asymptotic expansion for $\log(e^{-y}; e^{-\beta})_\infty$ as $\beta \to 0$. They utilize higher-order Stirling expansions involving Bernoulli numbers ($B_k$) and polylogarithms ($\text{Li}_s$).
• Scaling Regimes: The authors analyze the expansion under the scaling $y = x\beta^c$ for various $c \geq 0$, distinguishing between cases where the argument of the Gamma function is large, small, or of order unity.
• Error Analysis: In Section 5, the authors perform a heuristic error analysis (supported by numerical experiments in Mathematica) to determine the optimal truncation order ($N^*$) and the magnitude of the remainder ($R^*$) for the divergent asymptotic series.

3. Key Contributions and Results

A. The Main Product Formula

The central result is the identity (Equation 2 in the paper):
$$(e^{-y}; e^{-\beta})_\infty = \exp\left( \frac{\beta(1 - \coth(y/2))}{24} - \frac{\text{Li}_2(e^{-y})}{\beta} \right) (1 - e^{-y})^{1/2} \times \prod_{n \in \mathbb{Z}} \left( \frac{\sqrt{2\pi}}{\Gamma\left(\frac{y+2\pi i n}{\beta}\right)} \left(\frac{y+2\pi i n}{\beta}\right)^{\frac{y+2\pi i n}{\beta} - \frac{1}{2}} \exp\left( \frac{\beta}{12(y+2\pi i n)} - \frac{y+2\pi i n}{\beta} \right) \right)$$
This holds for $\text{Re}(\beta) > 0$ and $y \notin 2\pi i \mathbb{Z}$.

• Significance: This formula generalizes the modular transformation of Dedekind's eta function (recovered when $y=\beta$) to the non-modular $q$-Pochhammer symbol. It expresses the symbol as an infinite product of reciprocal Gamma functions "dressed" with exponential factors.

B. Uniform Asymptotic Expansion

The authors derive a uniform expansion for $\log(e^{-y}; e^{-\beta})_\infty$ (Corollary 2) valid in a specific region of the complex plane as $\beta \to 0$. The expansion includes terms involving polylogarithms $\text{Li}_{2-k}(e^{-y})$ and Bernoulli numbers.

C. Complete Asymptotics for Scaling Regimes

The paper provides explicit asymptotic series for $F(\beta) = \log(e^{-x\beta^c}; e^{-\beta})_\infty$ across four regimes (Corollary 3):

1. $c = 0$: The standard case where $y$ is fixed. The result recovers known expansions (related to Ramanujan's work) but provides a complete series form.
2. $0 < c < 1$: A new regime where the argument of the Gamma function grows as $\beta^{c-1}$. The expansion involves mixed powers of $\beta$ and logarithmic terms.
3. $c = 1$: The case where $y = x\beta$. This recovers the limit related to the Gamma function $\Gamma(x)$. The authors provide a complete expansion involving Bernoulli polynomials.
4. $c > 1$: A new regime where the argument of the Gamma function approaches a constant. The expansion involves Riemann zeta values $\zeta(k)$ and Euler's constant $\gamma$.

D. Connection to Narukawa's Identity and QFT

• Formal Limit: The authors show in Appendix B that their identity can be formally obtained as a limit ($p \to 0$) of Narukawa's identity for the elliptic gamma function.
• QFT Interpretation:
  • Narukawa's formula corresponds to the BPS partition function of a 4d $\mathcal{N}=1$ multiplet on $S^3 \times S^1$.
  • The new formula corresponds to the BPS partition function of a 3d $\mathcal{N}=2$ chiral multiplet on $D^2 \times S^1$.
  • The reduction from 4d to 3d is interpreted as a geometric degeneration of the manifold $S^3 \times S^1$ to $D^2 \times S^1$.

E. Error Analysis

The paper analyzes the divergence of the asymptotic series.

• The series are generally divergent (factorial growth of coefficients).
• The authors derive estimates for the optimal truncation order $N^*$ and the minimal error $R^*$.
• They find that for $c > 1$ or $c=1$ (with $x \notin \mathbb{Z}/2$), the error is dominated by the last sum in the product formula, behaving as $O(e^{-4\pi^2/\beta})$.
• For $0 \leq c < 1$, the dominant error comes from the Stirling expansion of the Gamma term, behaving as $O(e^{-2\pi x \beta^{c-1}})$.
• Numerical experiments confirm that the heuristic estimates for optimal truncation align well with computed errors.

4. Significance

• Mathematical Rigor: The paper provides a rigorous foundation for the asymptotic analysis of $q$-Pochhammer symbols in regimes previously only treated heuristically or partially.
• Resolution of Conjectures: The results are directly applicable to the conjecture in [ABF25] regarding the dominant scaling regimes in basic hypergeometric integrals, specifically addressing the critical cases $c \in \{0, 1/2, 1\}$.
• Physical Relevance: By linking the formula to the BPS partition functions of supersymmetric gauge theories on $D^2 \times S^1$, the work bridges the gap between special function theory and modern theoretical physics (specifically 3d/4d dualities and localization techniques).
• New Tools: The derived product formula serves as a powerful tool for analyzing modular transformations and limits of elliptic hypergeometric integrals, extending the utility of Narukawa's work to a broader class of functions.

In summary, this paper establishes a fundamental product identity for the $q$-Pochhammer symbol, leveraging it to solve long-standing problems in the asymptotic analysis of special functions and providing new insights into the structure of partition functions in quantum field theory.