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

Chen Wang, He-Xia Ni

TL;DR

The paper proves a parametric supercongruence linking Whipple's ${}_5F_4$ formula to Dwork's dash operation by introducing a new parametric WZ pair. It establishes a $p$-adic analogue for the case $a=b=c=\alpha$, $d=\tfrac12$, yielding a precise congruence for truncated hypergeometric sums modulo $p^{r+3}$ with a harmonic-number correction term. This result confirms Guo and Zhao's conjecture in a broad setting and extends the landscape of Ramanujan-type supercongruences through explicit dash-iterations and $p$-adic analysis. The methods provide a framework for parametric extensions of classical hypergeometric identities and highlight the role of Dwork's dash operation in congruence transformations, with potential connections to $p$-adic gamma values and related $p$-adic analytic structures.

Abstract

We establish a parametric supercongruence related to Whipple's ${}_5F_4$ formula and Dwork's dash operation. As a typical consequence, we obtain the following result: for any prime $p\equiv3\pmod4$ and odd integer $r\geq1$, $$ \sum_{k=0}^{p^r-1}(8k+1)\frac{(\frac14)_k^3(\frac12)_k}{(1)_k^3(\frac34)_k}\equiv 3p^r+\frac{27p^{3r}}{4}H_{(p^r-3)/4}^{(2)}\pmod{p^{r+3}}, $$ where $(x)_n=x(x+1)\cdots(x+n-1)$ is the Pochhammer symbol and $H_n^{(2)}=\sum_{k=1}^n\frac{1}{k^2}$ is the $n$-th harmonic number of order $2$. This confirms a conjecture of Guo and Zhao [Forum Math. 38 (2026), 1099-1109]. Our proof rely on a new parametric WZ pair which allows us to transform the original sum to a computable form in the sense of congruence. Another essential ingredient of our proof involves the properties of Dwork's dash operation.

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

TL;DR

The paper proves a parametric supercongruence linking Whipple's formula to Dwork's dash operation by introducing a new parametric WZ pair. It establishes a -adic analogue for the case , , yielding a precise congruence for truncated hypergeometric sums modulo with a harmonic-number correction term. This result confirms Guo and Zhao's conjecture in a broad setting and extends the landscape of Ramanujan-type supercongruences through explicit dash-iterations and -adic analysis. The methods provide a framework for parametric extensions of classical hypergeometric identities and highlight the role of Dwork's dash operation in congruence transformations, with potential connections to -adic gamma values and related -adic analytic structures.

Abstract

We establish a parametric supercongruence related to Whipple's formula and Dwork's dash operation. As a typical consequence, we obtain the following result: for any prime and odd integer , where is the Pochhammer symbol and is the -th harmonic number of order . This confirms a conjecture of Guo and Zhao [Forum Math. 38 (2026), 1099-1109]. Our proof rely on a new parametric WZ pair which allows us to transform the original sum to a computable form in the sense of congruence. Another essential ingredient of our proof involves the properties of Dwork's dash operation.
Paper Structure (3 sections, 13 theorems, 87 equations, 1 table)

This paper contains 3 sections, 13 theorems, 87 equations, 1 table.

Table of Contents

  1. Introduction
  2. Properties of Dwork's dash operation and immediate applications
  3. Proof of Theorem \ref{['mainth1']} and Corollary \ref{['cor']}

Key Result

Theorem 1.1

Let $c,d,s\in\mathbb Z^+$ with $d\geq 2$, $1\leq c,s\leq d$ and $\gcd(cs,d)=1$, and let $p\geq5$ be a prime with $p\equiv s\pmod d$. Then, for any $r\in\mathbb Z^+$ with $(\frac{1}{2}+\alpha)^{\ast_r}(\frac{1}{2}+\alpha^{\ast_r})\not\equiv0\pmod{p}$, we have where $\alpha=c/d$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.1
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • Corollary 2.2
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 12 more