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Vanishing Coefficients in Products of Quintuple Products

This paper utilizes involutive transformations on integer lattices to derive explicit arithmetic progressions modulo primes $p \equiv 1 \pmod{4}$ where the coefficients in the series expansions of specific products of quintuple products vanish.

Original authors: Taylor Daniels, Tim Huber, James McLaughlin, Dongxi Ye

Published 2026-06-08

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CC BY 4.0

Original authors: Taylor Daniels, Tim Huber, James McLaughlin, Dongxi Ye

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are looking at a giant, infinite musical score made of numbers. In this score, every note is a number (a coefficient) attached to a specific power of a variable called $q$ . Mathematicians call these "series expansions." Usually, when you play a song like this, you hear a constant stream of notes—numbers that go up, down, and around.

But sometimes, in very specific mathematical "songs," there are entire sections of silence. Certain notes simply never appear; their value is exactly zero. This paper is about finding the secret rules that tell us where these silences happen in a very complex type of musical score made by combining two special "quintuple product" instruments.

Here is a breakdown of what the authors, Taylor Daniels, Tim Huber, James McLaughlin, and Dongxi Ye, discovered, using simple analogies:

1. The Musical Instruments (The Quintuple Products)

Think of the "Quintuple Product" as a very special, complex instrument. When you play it, it produces a long list of numbers. The authors decided to play two of these instruments at the same time.

The Setup: They played one instrument tuned to a specific note ( $m$ ) and another tuned to a different note ( $n$ ).
The Condition: They only looked at instruments tuned to "prime numbers" (numbers like 5, 13, 17, 29) that have a special property: they can be written as the sum of two squares (like $13 = 2^2 + 3^2$ ).

2. The Mystery of the Missing Notes

When they combined the sounds of these two instruments, they noticed something strange. If you look at the notes at specific intervals (like every 13th note, or every 29th note), some of them are completely missing.

The Goal: They wanted to find a formula to predict exactly which notes would be missing.
The Discovery: They found that for any prime number $p$ (that fits their rules), there are specific "silence zones." If you count the notes in a certain pattern (an arithmetic progression), you will hit a note that is guaranteed to be zero.

3. The Secret Map (The "Involution")

How did they prove these notes were actually missing? They didn't just guess; they built a mathematical "magic mirror."

Imagine you have a giant grid of possibilities (a lattice). Every point on this grid represents a way the two instruments could combine to make a note.

The Mirror Trick: The authors found a special rule (a matrix transformation) that acts like a mirror. If you take any point on the grid and look at its reflection:
1. The reflection lands on a spot that produces the exact same note (same power of $q$ ).
2. However, the "volume" or "sign" of the note flips. If the original point was a positive note (+1), the reflection is a negative note (-1).
The Result: Because every note has a perfect twin that cancels it out (positive + negative = zero), the total sound for those specific notes becomes silence.

They used computers to test this mirror trick on many different prime numbers (like 13, 37, 61) until they could see the pattern and write down the general rule for the mirror.

4. The Two Types of Silence

The paper reveals that the "silence zones" depend on the type of prime number you are using:

Type A (Primes like 13, 37): These primes leave a remainder of 1 when divided by 12. Here, the silence happens in two specific patterns based on the numbers $m$ and $n$ used to tune the instruments.
Type B (Primes like 5, 29): These primes leave a remainder of 5 when divided by 12. Here, the silence patterns are slightly different, involving a bit more complex math with "inverse" numbers, but the result is the same: specific notes vanish.

5. Why Does This Matter?

The authors mention that while these products are usually "super-lacunary" (meaning they are mostly silence with only a few notes here and there), multiplying them by a specific "filter" (another mathematical function) changes the game.

The Filter: They showed that even if you add a "background hum" (a standard partition function) to the mix, the specific notes that were supposed to be silent stay silent.
The Implication: This helps mathematicians understand the deep structure of how numbers combine. It's like discovering that no matter how you rearrange a deck of cards, certain specific cards will always end up in the same hidden pockets.

Summary

In short, this paper is a detective story. The authors investigated a complex mathematical song made of two special instruments. They discovered that for certain prime numbers, specific notes in the song are guaranteed to be silent. They proved this by finding a "mirror rule" that pairs every positive note with a negative one, causing them to cancel each other out perfectly. They used computers to find the pattern and then wrote down the exact formula so anyone can predict where the silence will occur.

Technical Summary: Vanishing Coefficients in Products of Quintuple Products

Problem Statement
This paper investigates the phenomenon of vanishing coefficients in the series expansions of infinite $q$ -products, specifically focusing on products formed by two instances of the Quintuple Product Identity. The study extends previous work by Richmond and Szekeres, Andrews and Bressoud, and others, who identified arithmetic progressions modulo primes where coefficients in specific $q$ -series expansions vanish. The authors aim to derive explicit arithmetic progressions modulo primes $p \equiv 1 \pmod 4$ in which the coefficients of the product $Q(q^{bm}, q^p)Q(q^{bn}, q^p)$ are identically zero. Here, $p = m^2 + n^2$ , $b$ is a positive integer, and $Q(z, q)$ denotes the Quintuple Product.

Methodology
The authors employ a method fundamentally different from their previous work on this topic, which relied on $p$ -dissection and the appearance of infinite products of the form $(1; q^{p^2})_\infty$ . Instead, this paper utilizes involutive transformations on integer lattices.

The core strategy involves:

Formulation: Expressing the product $Q(q^{bm}, q^p)Q(q^{bn}, q^p)$ (multiplied by a non-vanishing factor $(q^{2p}; q^{2p})_\infty^2$ ) as a quadruple sum over integers $r, s, u, v$ using the Jacobi Triple Product Identity.
Restriction: Restricting the sum to terms where the exponent of $q$ is congruent to a specific value $b\tau \pmod p$ , where $\tau$ corresponds to the arithmetic progressions hypothesized to have vanishing coefficients.
Involution Construction: Defining a map $x \mapsto x' = Ax + b$ $x \mapsto x^{'} = A x + b$ on the integer lattice $\mathbb{Z}^4$ $Z^{4}$ (where $x = (r, s, u, v)$ $x = (r, s, u, v)$ ). The authors seek a matrix $A$ $A$ and vector $b$ $b$ such that:
- The map is an involution ( $A^2 = I$ and $(A+I)b = 0$ ).
- The exponent function $g(x)$ is invariant under the map ( $g(x') = g(x)$ ).
- The sign function $f(x)$ flips parity ( $f(x') \equiv f(x) + 1 \pmod 2$ ).
Numerical Exploration: The specific forms of $A$ and $b$ were initially discovered through extensive computer algebra experimentation (using Mathematica and PARI/GP) on various primes. The authors solved systems of quadratic equations derived from the condition $g(x') - g(x) = 0$ to identify patterns, which were then generalized and verified analytically.

Key Contributions and Results
The primary contribution is Theorem 3, which explicitly determines the arithmetic progressions where coefficients vanish for the sequence $\{a_t\}$ defined by:
$\sum_{t=0}^\infty a_t q^t = Q(q^{bm}, q^p)Q(q^{bn}, q^p)$
Let $w \equiv \overline{2}(m+n) \pmod p$ . The vanishing conditions depend on $p \pmod{12}$ :

Case 1 ( $p \equiv 1 \pmod{12}$ ):
The coefficients vanish for indices $t$ in the progressions:
$a_{pt + wb} = 0 \quad \text{and} \quad a_{pt + (w-3b)b} = 0$
(Note: The paper notation uses $apt+wb$ to denote $a_{pt+wb}$ ).
Case 2 ( $p \equiv 5 \pmod{12}$ ):
The coefficients vanish for indices $t$ in the progressions:
$a_{pt + (1-3b\overline{m})wb} = 0 \quad \text{and} \quad a_{pt + (1-3b\overline{n})wb} = 0$

The proof is completed by explicitly constructing the matrices $A$ and vectors $b$ for four distinct sub-cases (based on the divisibility of $m, n$ by 3 and the value of $p \pmod{12}$ ) and verifying that they satisfy the required involution properties.

Additionally, the paper presents Corollary 1, extending these vanishing results to the product divided by $(q^p; q^p)_\infty^2$ . This modification is significant because it relates the vanishing coefficients to partition functions, where the non-vanishing coefficients in other progressions are known to eventually maintain a constant sign.

Significance and Claims
The authors claim that their results provide a new, constructive proof for the vanishing of coefficients in these specific products, distinct from previous methods relying on $p$ -dissection. The paper emphasizes that the vanishing arises from symmetries in the expansions of products of two Quintuple Products.

The authors note that while the product of two superlacunary series (as defined by Ono and Robins) is lacunary (having many zero coefficients), their work identifies specific arithmetic progressions where these zeros occur systematically. They modestly state that their findings are derived from computer-assisted experimentation followed by rigorous algebraic verification.

In the concluding remarks, the authors suggest that similar vanishing phenomena may exist for products of three and four quintuple products, citing preliminary experimental data (e.g., for $p=31$ and $p=41$ ), but explicitly state that these cases will be examined in a subsequent paper, not within the scope of the current work.