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A further $q$ -generalization of the (C.2) and (G.2) supercongruences of Van Hamme

By applying the $q$ -Zeilberger algorithm, this paper establishes a unified $q$ -analogue of Van Hamme's (C.2) and (G.2) supercongruences, yielding new results including a $q$ -analogue involving Bernoulli numbers and a refinement of the (G.2) supercongruence.

Original authors: Song-Xiao Li, Su-Dan Wang

Published 2026-03-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Song-Xiao Li, Su-Dan Wang

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 a mathematician trying to solve a giant, cosmic puzzle. For decades, experts have been staring at a specific set of numbers that seem to follow a mysterious, hidden rhythm. These aren't just random numbers; they are supercongruences.

Think of a congruence like a clock. If you say "3 o'clock is congruent to 15 o'clock," you mean they look the same on the clock face. A supercongruence is like a super-accurate clock that doesn't just match the hour, but also the minutes, seconds, and even the milliseconds. It's a very strict rule that certain sums of numbers must follow when you divide them by a prime number (like 5, 7, 11, etc.).

The Story So Far

In 1997, a mathematician named Van Hamme found two of these super-strict rules (labeled C.2 and G.2). They were beautiful, but they were "static." They worked with standard numbers.

Later, other mathematicians asked: "What if we add a 'magic variable' called q to these rules?"
In the world of math, q is like a dimmer switch or a zoom lens.

When you set the dimmer to full brightness ( $q=1$ ), you see the original, standard numbers.
When you turn the dimmer down ( $q \neq 1$ ), the numbers stretch and warp into something called q-analogues. These are richer, more complex versions of the original rules.

Some people managed to turn the dimmer on for one of Van Hamme's rules, and others for the other rule. But nobody had found a single "universal remote" that could control both at the same time.

The New Discovery

The authors of this paper, Song-Xiao Li and Su-Dan Wang, have built that universal remote.

They used a powerful computer-assisted tool called the q-Zeilberger algorithm. Imagine this algorithm as a high-tech robotic chef. You give it a recipe (a complex mathematical formula), and it automatically chops, mixes, and tastes the ingredients until it finds a pattern that proves the recipe works.

Using this robotic chef, the authors created a Unified q-Analogue.

The Metaphor: Think of Van Hamme's two rules as two different types of musical instruments (say, a violin and a flute). Before this paper, we had sheet music for the violin and sheet music for the flute, but they were written in different languages.
The Breakthrough: Li and Wang wrote a new piece of sheet music that works for both instruments simultaneously. It's a single, elegant formula that, depending on how you tweak the "q" knob, transforms into either the violin piece or the flute piece.

Why Does This Matter?

You might ask, "Who cares about a dimmer switch for math formulas?"

It's a Deeper Truth: By finding the "q" version, the authors aren't just solving a puzzle; they are revealing the underlying DNA of these numbers. It shows that the rules Van Hamme found weren't accidents; they are just one specific setting of a much grander, more flexible law of the universe.
It Connects Distant Worlds: The paper shows that these number patterns are deeply connected to Bernoulli numbers (famous numbers used in calculus and physics) and Gamma functions (tools used to calculate factorials for non-whole numbers). The new formula acts as a bridge, allowing mathematicians to walk from the world of simple integers to the world of complex analysis more easily.
It's Stronger: The authors didn't just find a new formula; they proved it works with even higher precision than anyone expected. They strengthened the "super" in supercongruence, proving the rules hold true even when you look at them under a microscope (modulo $p^5$ instead of just $p^3$ ).

The Takeaway

In simple terms, Li and Wang took two separate, mysterious mathematical riddles and realized they were actually two sides of the same coin. They built a master key (the unified q-analogue) that unlocks both riddles at once, revealing a deeper, more beautiful structure in the fabric of numbers.

They didn't just solve the puzzle; they showed us that the puzzle pieces were part of a much larger, more magnificent picture all along.

1. Problem Statement

The paper addresses the problem of establishing unified $q$ -analogues for two specific supercongruences originally proposed by Van Hamme in 1997, labeled (C.2) and (G.2). These supercongruences are $p$ -adic analogues of Ramanujan-type series for $1/\pi$ .

Van Hamme's (C.2):
$\sum_{k=0}^{(p-1)/2} (4k+1) \frac{(\frac{1}{4})_k^4}{k!^4} \equiv p \pmod{p^3}$
Van Hamme's (G.2):
$\sum_{k=0}^{(p-1)/4} (8k+1) \frac{(\frac{1}{2})_k^4}{k!^4} \equiv p \frac{\Gamma_p(1/2)\Gamma_p(1/4)}{\Gamma_p(3/4)} \pmod{p^3}$

While previous works (by Long, Guo, Wang, He, Liu, and others) had established $q$ -analogues for specific cases or modulo lower powers (typically modulo $[n]\Phi_n(q)^3$ ), this paper aims to provide a more general and refined $q$ -supercongruence that holds modulo $[n]\Phi_n(q)^4$ . This higher modulus represents a significant strengthening of existing results, allowing for the derivation of new congruences involving Bernoulli numbers and higher powers of primes ( $p^5$ ) upon taking the limit $q \to 1$ .

2. Methodology

The authors employ a combination of advanced techniques from the theory of basic hypergeometric series and $q$ -series:

The $q$ -Zeilberger Algorithm: The core of the proof relies on constructing a specific telescoping pair of rational functions, $F(m, k)$ and $G(m, k)$ , which satisfy a recurrence relation of the form:
$[dk - d + r]F(m, k-1) - [dk - d + 2r]F(m, k) = G(m+1, k) - G(m, k)$
Summing this relation allows the transformation of a complex summation into a boundary term and a residual sum.
Cyclotomic Polynomials and Modulo Arithmetic: The proofs are conducted modulo $[n]\Phi_n(q)^4$ , where $[n] = \frac{1-q^n}{1-q}$ is the $q$ -integer and $\Phi_n(q)$ is the $n$ -th cyclotomic polynomial. This requires precise handling of $q$ -congruences involving roots of unity.
Polynomial Coprimality Lemmas: A significant portion of the technical work involves proving that certain denominators in the $q$ -hypergeometric terms are coprime to $1-q^n$ and $\Phi_n(q)$ . The authors establish several lemmas (Lemmas 2.1–2.6 and 4.1–4.4) to ensure that divisions are valid within the specified modulus.
Limiting Processes: To connect the $q$ -results back to classical number theory, the authors analyze the behavior of the $q$ -supercongruences as $q \to 1$ and set $n = p$ (a prime). They utilize known congruences for harmonic numbers and Bernoulli numbers (specifically results by Z.-W. Sun and Z.-H. Sun) to verify the consistency of their $q$ -analogues with classical results.

3. Key Contributions and Results

A. Unified $q$ -Generalization (Theorem 1.1)

The primary contribution is Theorem 1.1, which establishes a unified $q$ -supercongruence for integers $n, d$ with $\gcd(n,d)=1$ and $n \equiv r \pmod d$ .

Modulus: The result holds modulo $[n]\Phi_n(q)^4$ .
Formula: It generalizes previous results (such as Guo's unified form modulo $[n]\Phi_n(q)^3$ ) by including higher-order terms involving $[n]^3$ and $[n]^4$ .
Structure: The sum involves terms like $[2dk+r] \frac{(q^r; q^d)_k^4}{(q^d; q^d)_k^4} q^{(d-2r)k}$ . The right-hand side includes a main term and correction terms involving sums of $q$ -harmonic-like series.

B. Refinement of (C.2) (Corollary 1.2)

By setting $d=2$ and $r=1$ in Theorem 1.1, the authors derive a $q$ -analogue of the (C.2) supercongruence.

Significance: Taking $q \to 1$ and $n=p$ recovers the classical supercongruence modulo $p^4$ (Long's result) and further refines it to include terms related to Bernoulli numbers $B_{p-3}$ , effectively proving a partial $q$ -analogue of a conjecture by Guo.

C. Refinement of (G.2) (Theorem 1.4 and Corollary 1.5)

The paper provides a second major theorem (Theorem 1.4) for the case where $n \equiv -r \pmod d$ , which corresponds to the (G.2) type supercongruences.

Result: This leads to Corollary 1.5, a $q$ -supercongruence for $p \equiv 3 \pmod 4$ .
Advancement: By letting $q \to 1$ , the authors strengthen the known (G.2) supercongruence (previously modulo $p^4$ ) to modulo $p^5$ .
New Identity: This process yields Corollary 1.6, a new classical congruence involving the $p$ -adic Gamma function and harmonic numbers:
$\dots \equiv (-1)^{(p-3)/4} 3^2 p^2 \Gamma_p(1/4)^2 \Gamma_p(1/2) \pmod{p^4}$

4. Significance

Unification: The paper successfully unifies two distinct families of supercongruences ((C.2) and (G.2)) under a single $q$ -framework, demonstrating a deeper structural connection between them.
Higher Precision: By pushing the modulus from $[n]\Phi_n(q)^3$ to $[n]\Phi_n(q)^4$ , the authors capture more subtle arithmetic information. This allows for the derivation of congruences modulo $p^5$ in the classical limit, which were previously unknown or unproven.
Methodological Rigor: The paper provides a robust template for proving higher-order $q$ -supercongruences using the $q$ -Zeilberger algorithm combined with careful analysis of cyclotomic polynomial properties.
New Conjectures and Identities: The results verify existing conjectures (e.g., by Guo) and generate new identities involving Bernoulli numbers and $p$ -adic Gamma functions, enriching the landscape of $p$ -adic analysis and combinatorial number theory.

In summary, Li and Wang have significantly advanced the field of $q$ -supercongruences by providing a unified, higher-order generalization of Van Hamme's conjectures, bridging the gap between $q$ -series theory and classical $p$ -adic congruences with unprecedented precision.