## Abstract

We prove some supercongruences involving the Apéry polynomials

$${A}_{n}(x)=\sum _{k=0}^{n}{\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}^{\phantom{\rule{-0.17em}{0ex}}2}{\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right)}^{\phantom{\rule{-0.17em}{0ex}}2}{x}^{k}(n\in \mathbb{N}=\{0,1,\dots ,\}),$$

the generalized Domb numbers

$${D}_{n}(A,B,C)=\sum _{k=0}^{n}{\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}^{\phantom{\rule{-0.17em}{0ex}}A}{\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}^{\phantom{\rule{-0.17em}{0ex}}B}{\left(\genfrac{}{}{0.0pt}{}{2n-2k}{n-k}\right)}^{\phantom{\rule{-0.17em}{0ex}}C}(n\in \mathbb{N})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{Q}_{n}=\sum _{k=0}^{n}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{n-k}{k}\right)\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right)(n\in \mathbb{N}),$$

which were conjectured by Z.-W. Sun. For example, we show that for any prime $p>3$ and positive integer $r$ we have

$$\frac{{A}_{{p}^{r}}(-1)-{A}_{{p}^{r-1}}(-1)}{{p}^{3r}}\equiv \frac{29}{6}{B}_{p-3}\phantom{\rule{0.3em}{0ex}}(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}p)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{{Q}_{{p}^{r}}-{Q}_{{p}^{r-1}}}{{p}^{3r}}\equiv -\frac{1}{9}{B}_{p-3}\phantom{\rule{0.3em}{0ex}}(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}p),$$

where ${B}_{0},{B}_{1},{B}_{2},\dots $ are the Bernoulli numbers. The following supercongruences hold modulo $p$:

$$\frac{{D}_{{p}^{r}}(A,1,1)-{D}_{{p}^{r-1}}(A,1,1)}{{p}^{(A+1)r}}\equiv \{\begin{array}{cc}8\left(\frac{-1}{{p}^{r}}\right){E}_{p-3},\phantom{\rule{1em}{0ex}}\hfill & \text{if}A=1,\hfill \\ & \\ \frac{16}{3}{B}_{p-3},\phantom{\rule{1em}{0ex}}\hfill & \text{if}A=2,\hfill \end{array}$$

where $\left(\frac{\cdot}{p}\right)$ denotes the Legendre symbol and ${E}_{0},{E}_{1},{E}_{2},\dots $ are the Euler numbers.

## Citation

Yong Zhang. "Some conjectural supercongruences related to Bernoulli and Euler numbers." Rocky Mountain J. Math. 52 (3) 1105 - 1126, June 2022. https://doi.org/10.1216/rmj.2022.52.1105

## Information