He-Xia Ni

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.

Let $p$ be an odd prime and $x$ be an indeterminate. Recently, Z.-W. Sun proposed the following conjecture: $$\det\left[x+\left(\frac{j-i}{p}\right)\right]_{0\le i,j\le \frac{p-1}{2}}=\begin{cases} (\frac{2}{p})pb_px-a_p & \mbox{if}\ p\equiv 1\pmod4, 1 & \mbox{if}\ p\equiv 3\pmod4, \end{cases}$$ where $a_p$ and $b_p$ are rational numbers related to the fundamental unit and class number of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In this paper, we confirm the above conjecture of Sun based on Vsemirnov's decomposition of Chapman's "evil determinant".