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A supercongruence related to Whipple's ${}_5F_4$ formula and Dwork's dash operation

This paper establishes a parametric supercongruence related to Whipple's ${}_5F_4$ formula and Dwork's dash operation, thereby confirming a conjecture by Guo and Zhao regarding a specific summation involving Pochhammer symbols and harmonic numbers modulo $p^{r+3}$ .

Original authors: Chen Wang, He-Xia Ni

Published 2026-02-16

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CC BY 4.0

Original authors: Chen Wang, He-Xia Ni

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 detective trying to solve a mystery involving a very strange, infinite recipe. This recipe is a mathematical formula discovered by the legendary Indian genius Srinivasa Ramanujan over a century ago. It's a recipe for calculating numbers related to $\pi$ (the ratio of a circle's circumference to its diameter) using an endless list of ingredients.

For a long time, mathematicians knew this recipe worked perfectly when you added up all the ingredients (infinity). But what happens if you stop the recipe early? What if you only use the first few ingredients?

The Mystery: Truncated Recipes and "Super" Patterns

In the 1990s, a mathematician named Van Hamme noticed something weird. When you stop Ramanujan's recipe at a specific point (related to a prime number, like 5, 7, or 11), the result isn't just a random number. It follows a hidden, ultra-precise pattern called a supercongruence.

Think of a normal congruence like a clock. If it's 10 o'clock and you add 4 hours, it's 2 o'clock. The "remainder" is 2.
A supercongruence is like a clock that is so precise it can tell you not just the hour, but the exact second, the exact millisecond, and even the exact nanosecond, all at once. It's a pattern that holds true with incredible accuracy, far beyond what you'd expect.

The New Clue: A Conjecture

Recently, two other mathematicians, Guo and Zhao, looked at a specific variation of this recipe (involving the number 1/4 and 1/2). They found a pattern that worked for many cases, but they hit a wall. They guessed that the pattern should hold true even with even more precision (a "modulus" of $p^{r+3}$ ), but they couldn't prove it. It was like seeing a footprint in the mud and guessing the shoe size, but not being able to find the shoe to confirm it.

The Solution: Wang and Ni's Detective Work

In this paper, Chen Wang and He-Xia Ni act as the detectives who finally find the shoe. They prove that Guo and Zhao's guess was correct.

Here is how they did it, using some creative metaphors:

1. The "Dash" Operation (The Magic Shrink Ray)
The paper uses a tool called Dwork's dash operation. Imagine you have a number, and you want to see its "core" structure. The dash operation is like a magic shrink ray. It takes a number, strips away the messy parts, and reveals a simpler, underlying version of itself. The authors use this to simplify the complex ingredients of the recipe, making them easier to handle.

2. The WZ Pair (The Perfectly Balanced Scale)
The biggest hurdle was that the ingredients in the recipe were messy. Some were whole numbers, and some were fractions that didn't behave nicely. To fix this, the authors found a WZ pair.
Think of a WZ pair as a set of perfectly balanced scales. You have a "Left Side" (the messy sum you want to calculate) and a "Right Side" (a much simpler, computable sum). The magic of the WZ pair is that if you know the value of the Right Side, you automatically know the value of the Left Side, even if the Left Side looks impossible to calculate directly.
The authors built a new type of scale specifically for this problem, allowing them to transform the impossible sum into a manageable one.

3. The Result: Confirming the Conjecture
By using their new scale and the magic shrink ray, they managed to calculate the sum exactly. They proved that for any prime number $p$ (that leaves a remainder of 3 when divided by 4) and any odd number $r$ , the sum follows this specific, ultra-precise pattern:

$\text{Sum} \approx 3p^r + \text{a tiny correction term}$

This "tiny correction term" involves a harmonic number (a sum of fractions like $1 + 1/4 + 1/9 + \dots$ ), which acts like the fine-tuning knob on a radio to get the signal perfectly clear.

Why Does This Matter?

You might ask, "Who cares about these weird number patterns?"

Mathematical Beauty: It connects different areas of math. The paper links hypergeometric series (complex infinite sums), p-adic numbers (a weird way of measuring distance in numbers), and Gamma functions (generalized factorials). It's like discovering that a song written in the 1920s shares the exact same DNA as a song written in 2026.
Solving Old Puzzles: It confirms a conjecture that was sitting on a shelf, waiting for someone to solve it.
New Tools: The "new parametric WZ pair" they invented is a new tool in the mathematician's toolbox. Other researchers can now use this tool to solve different, perhaps even harder, mysteries in number theory.

In a Nutshell

Wang and Ni took a complex, infinite mathematical recipe, stopped it at a specific point, and proved that the result follows a hidden, ultra-precise pattern. They did this by inventing a new mathematical "balance scale" (WZ pair) and using a "shrink ray" (Dwork's dash) to simplify the ingredients. Their work confirms a guess made by colleagues and opens the door for future discoveries in the world of numbers.

1. Problem Statement and Motivation

The paper addresses the problem of establishing parametric supercongruences for truncated hypergeometric series, specifically those related to Ramanujan-type series for $1/\pi$ .

Context: Ramanujan discovered series for $1/\pi$ involving hypergeometric terms. Van Hamme later conjectured $p$ -adic analogues (supercongruences) for these series modulo powers of primes.
Specific Gap: Guo and Zhao (2026) recently studied $q$ -supercongruences and conjectured a refined supercongruence for the case $p \equiv 3 \pmod 4$ . Specifically, they conjectured that for an odd integer $r \ge 1$ :
$\sum_{k=0}^{p^r-1} \frac{(8k+1)(\frac{1}{4})_k^3 (\frac{1}{2})_k}{(1)_k^3 (\frac{3}{4})_k} \equiv 3p^r + \frac{27}{4}p^{3r} H_{(p^r-3)/4}^{(2)} \pmod{p^{r+3}}$
where $H_n^{(2)}$ is the harmonic number of order 2.
Goal: The authors aim to prove this conjecture and, more broadly, establish a general parametric supercongruence related to Whipple's ${}_5F_4$ summation formula using Dwork's dash operation.

2. Methodology

The authors employ a combination of advanced techniques from $p$ -adic analysis and combinatorial hypergeometric theory:

Dwork's Dash Operation:
- They utilize the operation $x^* = (x + \langle -x \rangle_p)/p$ , where $\langle \cdot \rangle_p$ denotes the least non-negative residue modulo $p$ .
- They define iterated operations $x^{*r}$ and analyze their properties, particularly how they relate to the $p$ -adic valuation of Pochhammer symbols $(x)_n$ .
- Key lemmas establish exact values for $\alpha^{*r}$ and the behavior of $(1/2 + \alpha)_l / (1/2 + \alpha + p^r)_l$ modulo $p$ .
Creative Microscoping and WZ Pairs:
- Instead of directly applying Whipple's infinite series formula (which fails due to non-integrality of summands in the $p$ -adic sense), the authors construct a new parametric WZ (Wilf-Zeilberger) pair $(F, G)$ .
- The pair is defined as:
  $F(x, k) = \frac{(2k+x)(x)_k^3 (1/2)_k}{(1)_k^3 (1/2+x)_k}, \quad G(x, k) = \frac{k^3(k+2x)}{x^3} F(x, k)$
- They verify the WZ relation: $F(x+1, k) - F(x, k) = G(x, k+1) - G(x, k)$ .
- This allows them to transform the original sum $\sum F(\alpha, k)$ into a telescoping sum involving $G$ , which is computable in the sense of congruence.
$p$ -adic Gamma Function and Harmonic Number Reductions:
- The proof relies on properties of the Morita $p$ -adic Gamma function $\Gamma_p(x)$ and specific reductions of harmonic numbers $H_n^{(2)}$ modulo powers of $p$ .
- They establish lemmas showing that certain sums of inverse squares vanish modulo $p^3$ under specific conditions.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

The authors establish a general parametric supercongruence. Let $c, d, s \in \mathbb{Z}^+$ with $d \ge 2$ , $1 \le c, s \le d$ , $\gcd(cs, d)=1$ , and $p \ge 5$ be a prime such that $p \equiv s \pmod d$ . Let $\alpha = c/d$ . For any $r \in \mathbb{Z}^+$ satisfying $(1/2 + \alpha)^{*r}(1/2 + \alpha^{*r}) \not\equiv 0 \pmod p$ :

$\sum_{k=0}^{p^r-1} \frac{(2k+\alpha)(\alpha)_k^3 (1/2)_k}{(1)_k^3 (1/2+\alpha)_k} \equiv \alpha^{*r} p^r - \frac{(\alpha^{*r})^3}{(1/2+\alpha)^{*r}} p^{r+2} H_{\alpha^{*r}p - \alpha^{*r} - 1}^{(2)} \pmod{p^{r+3}}$

Significance: This is a $p$ -adic analogue of Whipple's ${}_5F_4$ formula. It handles cases where individual summands are not $p$ -adic integers, yet their sum is, by utilizing the WZ pair to bypass direct term-wise analysis.

Confirmation of Conjecture (Corollary 1.1)

By setting specific parameters ( $d=4, s=3, c=1$ ) and requiring $r$ to be odd, the authors prove Guo and Zhao's conjecture:
$\sum_{k=0}^{p^r-1} \frac{(8k+1)(\frac{1}{4})_k^3 (\frac{1}{2})_k}{(1)_k^3 (\frac{3}{4})_k} \equiv 3p^r + \frac{27}{4}p^{3r} H_{(p^r-3)/4}^{(2)} \pmod{p^{r+3}}$
This holds for any prime $p \equiv 3 \pmod 4$ .

Additional Result

The paper also derives a stronger version of Van Hamme's supercongruence (C.2) for $p \equiv 1 \pmod 4$ (specifically for the case $d=2, s=1, c=1$ ):
$\sum_{k=0}^{p^r-1} \frac{(4k+1)(\frac{1}{2})_k^4}{(1)_k^4} \equiv p^r \pmod{p^{r+3}}$

4. Significance

Resolution of Open Problems: The paper definitively confirms a recent conjecture by Guo and Zhao regarding the modulus $p^{r+3}$ for a specific Ramanujan-type series, extending previous results that were only known modulo $p^{r+2}$ .
Methodological Innovation: The use of a parametric WZ pair to handle sums where terms are not $p$ -adic integers is a significant technical advancement. It overcomes the limitations of applying classical Whipple formulas directly in the $p$ -adic setting.
Unification: The results unify various known supercongruences under a single parametric framework involving Dwork's dash operation, providing a deeper structural understanding of the relationship between hypergeometric series and $p$ -adic arithmetic.
Harmonic Number Identities: The proof introduces new identities relating truncated harmonic numbers of order 2 to $p$ -adic parameters, which may have applications in other areas of number theory.

In summary, this paper provides a rigorous proof of a high-order supercongruence conjecture by developing a novel combination of WZ theory and $p$ -adic analysis, thereby advancing the understanding of $p$ -adic analogues of classical hypergeometric identities.