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

TL;DR

This paper proves a new supercongruence related to a hypergeometric series and Dwork's dash operation, confirming a recent conjecture and introducing a novel WZ pair for the proof.

Contribution

It establishes a parametric supercongruence linked to Whipple's formula, confirming a conjecture and utilizing a new WZ pair and Dwork's dash operation properties.

Findings

Proved a supercongruence for primes congruent to 3 mod 4.

Confirmed a conjecture of Guo and Zhao from 2026.

Developed a new parametric WZ pair for sum transformation.

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.