A supercongruence related to Whipple's ${}_5{F}_4$ formula and Dwork's dash operation

Abstract

We establish a parametric supercongruence related to Whipple's ${}_5{F}_4$ formula and Dwork's dash operation. As a typical consequence, we obtain the following result: for any prime $p\equiv 3(\mathrm{mod}4)$ and odd integer $r\ge 1$, $$\sum _{k=0}^{p^r-1}(8k+1)\frac{(\frac{1}{4}{)}_k^3(\frac{1}{2}{)}_k}{(1{)}_k^3(\frac{3}{4}{)}_k}\equiv 3p^r+\frac{27p^{3r}}{4}{H}_{(p^r-3)/4}^{(2)}(\mathrm{mod}{p}^{r+3}),$$ 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.

16 pages