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Supercongruences Arising from Truncated Appell Series

This paper investigates the congruence properties of truncated Appell and $q$ -Appell series by employing combinatorial identities and the creative microscoping method to establish $q$ -analogues for known results, prove a new $q$ -supercongruence for the $F_4$ series, and propose a conjecture extending the work of Apagodu and Zeilberger.

Original authors: He-Xia Ni

Published 2026-07-13

📖 6 min read🧠 Deep dive

CC0 1.0

Original authors: He-Xia Ni

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 the world of numbers as a giant, bustling city where every building is a mathematical formula. For a long time, mathematicians have been studying a specific type of "double-decker" building called an Appell series. Think of these as two-dimensional towers built by stacking blocks in a grid pattern, where the height of the tower depends on two variables, $x$ and $y$ .

In recent years, researchers have started looking at truncated versions of these towers. Instead of building them all the way to the infinite sky, they stop the construction at a specific floor, say floor $n$ . This creates a finite, manageable structure that is much easier to inspect. The big question mathematicians have been asking is: "If we look at these truncated towers through a special magnifying glass called a modulus (specifically looking at remainders when divided by prime numbers), do they reveal hidden patterns or 'super-congruences'?"

This paper, written by He-Xia Ni, dives deep into these truncated towers, but with a twist. The author doesn't just look at the standard buildings; they investigate q-analogues. You can think of a "q-analogue" as a magical version of the building where the bricks themselves are slightly different—they have a "quantum" flavor (represented by the letter $q$ ) that changes how they fit together. When $q$ is set to 1, the magic fades, and you get the ordinary building back.

The Main Discovery: A New Set of Blueprints

The paper's primary achievement is providing a new set of blueprints (formulas) that predict exactly what these magical, truncated towers look like when viewed through the lens of prime numbers.

Specifically, the author focuses on four types of these towers, labeled $F_1$ , $F_2$ , $F_3$ , and $F_4$ . The paper proves that for certain types of these towers, when you stop the construction at a specific floor related to a prime number $p$ , the total number of blocks follows a very strict, predictable rule modulo $p^2$ (meaning the remainder when divided by $p^2$ is fixed).

For instance, the paper confirms a "q-analogue" for a famous conjecture about the $F_4$ tower. Imagine a previous guess that said, "If you build this tower up to floor $p-1$ , the total blocks will leave a specific remainder." This paper proves that the magical version of that tower (the q-analogue) follows a similar, but more complex, rule involving the cyclotomic polynomial $\Phi_n(q)$ (a special mathematical tool that acts like a sieve for roots of unity).

The "Creative Microscope" Tool

How did the author find these patterns? They used a technique called "creative microscoping," a method introduced by Guo and Zudilin. You can imagine this as a super-powerful microscope that doesn't just zoom in on the blocks, but allows the mathematician to tweak the parameters of the building (like changing the size of the bricks or the angle of the walls) to see how the structure behaves under extreme pressure. By using this microscope alongside some classic combinatorial identities (mathematical recipes for counting), the author was able to derive these new formulas.

What Was Proven vs. What Was Suggested

It is important to note exactly how sure the author is about these findings:

Proven Facts: The paper provides rigorous, step-by-step proofs for several new formulas. For example, Theorem 1.1 and Theorem 1.3 are presented as solid mathematical facts. The author proves that for any positive integer $n$ (with specific conditions), the truncated series behaves in a certain way modulo $\Phi_n(q)^2$ . These are not guesses; they are derived using established mathematical logic, including the Chinese Remainder Theorem and properties of roots of unity.
Conjectures (Suggested Ideas): The paper also touches on a "Conjecture 3.1." Here, the author suggests that a similar pattern might hold for a specific case where $n \equiv 3 \pmod 4$ , but they state this is based on "comprehensive numerical evidence" rather than a full proof. In this case, the paper is saying, "Our computer simulations and calculations strongly suggest this is true, but we haven't written the final proof yet."
What Was Ruled Out: The paper does not explicitly argue against a specific rival theory in a hostile way, but it does clarify that a previous attempt to prove a specific result (Theorem 1.4) using only the "creative microscoping" method failed. The author admits they couldn't prove the numerator contained high enough powers of the polynomial using that method alone. Instead, they had to switch tactics, bringing in q-differential operators (tools that measure how the magical bricks change as you tweak $q$ ) and properties of q-roots of unity to succeed. This isn't a rejection of the old method, but a demonstration that a different tool was necessary for this specific job.

The "Super" in Supercongruence

Why call them "supercongruences"? In the world of math, a regular congruence is like saying two numbers leave the same remainder when divided by 5. A "supercongruence" is a much stronger statement, often involving higher powers like $p^2$ or $p^3$ . It's like saying not only do the buildings look the same from the street, but if you count every single brick in the foundation, they match perfectly down to the last grain of sand.

The paper establishes that these truncated Appell series, when viewed through the q-magic lens, obey these incredibly strong rules. For example, Theorem 1.4 shows a relationship between a large tower built up to $rn-1$ floors and a smaller tower built up to $r-1$ floors, linking them with a specific power of $q$ and a Legendre symbol (a mathematical switch that tells you if a number is a "square" or not).

In Summary

He-Xia Ni's paper is a tour de force in the world of number theory. It takes the complex, double-layered structures of Appell series, shrinks them down to finite sizes, and applies a magical "q" filter. Using a mix of high-powered mathematical microscopes and new tools like differential operators, the author proves that these structures follow elegant, predictable patterns when divided by prime numbers. While some parts of the map are still marked with "suggested" based on strong numerical evidence, the core findings are solid, proven facts that add new, vibrant colors to our understanding of how these mathematical towers stand.

Technical Summary: Supercongruences Arising from Truncated Appell Series

Problem Statement
This paper investigates the congruence properties of truncated Appell series ( $F_1, F_2, F_3, F_4$ ) and their $q$ -analogues (truncated $q$ -Appell series $\Phi^{(1)}, \Phi^{(2)}, \Phi^{(3)}, \Phi^{(4)}$ ). While recent literature has established various supercongruences for these series modulo prime powers (e.g., $p^2$ ), there is a need for corresponding $q$ -analogues modulo cyclotomic polynomials ( $\Phi_n(q)$ ) and their squares. Specifically, the paper addresses the lack of $q$ -analogues for known congruences involving $F_1$ and $F_2$ , and seeks to establish new $q$ -supercongruences for $F_4$ that relate to conjectures by Apagodu and Zeilberger.

Methodology
The author employs a combination of combinatorial identities and the "creative microscoping" method, originally introduced by Guo and Zudilin. The core technical approach involves:

Creative Microscoping: Introducing auxiliary parameters (e.g., $a, b$ ) into the series to create congruences modulo factors like $(a-q^n)$ and $(b-q^n)$ . The Chinese Remainder Theorem is then applied to combine these results.
$q$ -Hypergeometric Transformations: Utilizing reduction formulas for basic hypergeometric series, specifically the transformation of $\Phi^{(1)}$ into a ${}_2\phi_1$ series (Eq. 1.16).
Summation Formulas: Extensive use of the $q$ -Vandermonde (or $q$ -Chu-Vandermonde) summation formula (Eq. 1.17, 1.18) and Andrews' terminating $q$ -analogue of Watson's ${}_3F_2$ sum (Eq. 1.20).
Roots of Unity and Differential Operators: For more complex cases (specifically Theorem 1.4), the author utilizes properties of $q$ -roots of unity and $q$ -differential operators to establish crucial identities, as direct application of creative microscoping proved insufficient for certain numerator structures.
Limiting Cases: Deriving classical supercongruences (modulo $p^2$ ) by setting $n=p$ (an odd prime) and taking the limit $q \to 1$ .

Key Contributions and Results

$q$ -Analogues for $F_1$ (Theorem 1.1):
The paper provides $q$ -analogues for the congruence $F_1[\frac{1}{2}; \frac{1}{2}, \frac{1}{2}; 1; 1, 1]_{\frac{p-1}{2}} \equiv p \pmod{p^2}$ .
- For $d=2, r=1$ , a specific $q$ -congruence is established modulo $[n]\Phi_n(q)$ .
- For general $d \ge 2$ , a complex $q$ -congruence is derived modulo $\Phi_n(q)^2$ involving harmonic-like sums.
- A new identity (Eq. 1.12) is presented for a double sum related to these series.
- Corollary 1.1 recovers the classical supercongruences for $F_1$ by taking limits.
$q$ -Analogues for $F_2$ and $F_4$ (Theorems 1.2 & 1.3):
- Theorem 1.2 establishes a $q$ -congruence modulo $\Phi_n(q)$ for a truncated series related to $F_2$ . The result depends on $n \pmod 4$ : it vanishes if $n \equiv 3 \pmod 4$ and yields a specific product of $q$ -shifted factorials if $n \equiv 1 \pmod 4$ .
- Theorem 1.3 provides a stronger result modulo $\Phi_n(q)^2$ for a series related to $F_4$ . This leads to Corollary 1.2, which confirms the supercongruence for $F_4[\frac{1}{2}, \frac{1}{2}; 1, 1; \frac{1}{4}, \frac{1}{4}]_{\frac{p-1}{2}}$ modulo $p^2$ , distinguishing cases based on $p \pmod 4$ .
Resolution of Apagodu-Zeilberger Conjectures (Theorem 1.4):
The paper addresses Super-Conjecture 5' by Apagodu and Zeilberger regarding the ratio of truncated series $F_4[1, 1; 1, 1; 1, 1]$ .
- Theorem 1.4 proves a $q$ -supercongruence modulo $\Phi_n(q)^2$ relating the truncated series of length $rn$ and $sn$ to those of length $r$ and $s$ .
- Corollary 1.2 (in the context of Theorem 1.4) demonstrates that setting $n=p$ and $q \to 1$ recovers the conjectured classical congruence: $F_4[1, 1; 1, 1; 1, 1]_{rp-1} \equiv (\frac{p}{3}) F_4[1, 1; 1, 1; 1, 1]_{r-1} \pmod{p^2}$ .
Conjecture:
Based on numerical evidence, the author proposes Conjecture 3.1, suggesting that the series in Theorem 1.2 vanishes modulo $\Phi_n(q)^2$ when $n \equiv 3 \pmod 4$ .

Significance and Claims
The paper claims to provide the first $q$ -analogues for several known supercongruences of truncated Appell series $F_1$ and $F_2$ . Furthermore, it establishes a new $q$ -supercongruence for the truncated $q$ -Appell series $\Phi^{(4)}$ (related to $F_4$ ), which serves as a $q$ -analogue for two specific conjectures by Apagodu and Zeilberger. The work extends the understanding of the arithmetic properties of these double hypergeometric series by lifting classical results to the $q$ -setting, utilizing advanced techniques involving cyclotomic polynomials and creative microscoping. The results are presented as rigorous proofs of these $q$ -congruences, with the classical cases emerging naturally as limiting values.

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