Some conjectural supercongruences related to Bernoulli and Euler numbers. (English) Zbl 1497.11013

Zhang, Yong

Rocky Mt. J. Math. 52, No. 3, 1105-1126 (2022).

Summary: We prove some supercongruences involving the Apéry polynomials \[A_n (x) = \sum^n_{k = 0} \binom{n}{k}^2\binom{n+k}{k}^2 x^k \,\, (n\in\mathbb{N}=\{0, 1, \ldots,\}),\] the generalized Domb numbers \[D_n (A, B, C) = \sum^n_{k = 0} \binom{n}{k}^A\binom{2k}{k}^B\binom{2n-2k}{n-k}^C\,\, (n\in\mathbb{N})\,\,\text{and}\,\, Q_n=\sum^n_{k=0}\binom{n}{k}\binom{n-k}{k}\binom{n+k}{k}\,\,(n\in\mathbb{N}),\] which were conjectured by Z.-W. Sun. For example, we show that for any prime \(p > 3\) and positive integer \(r\) we have \[\frac{A_{p^{r}} (-1) - A_{p^{r - 1}} (-1)} {p^{3r}} \equiv \frac{29}{6}B_{p - 3} \;(\mathrm{mod}\;p)\,\,\text{and}\,\,\frac{Q_{p^{r}} - Q_{p^{r - 1}}} {p^{3r}} \equiv - \frac{1}{9}B_{p - 3} \; (\mathrm{mod}\;p),\] where \(B_0, B_1, B_2, \ldots\) are the Bernoulli numbers. The following supercongruences hold modulo \(p\):
\[\frac{D_{p^{r}} (A, 1, 1) - D_{p^{r - 1}} (A, 1, 1)} {p^{(A + 1)r}} \equiv \begin{cases} 8(\frac{-1}{p^{r}})E_{p-3}, & \text{if } A=1 \\ \frac{16}{3}B_{p-3}, & \text{if } A=2, \end{cases}\] where \((\frac{\cdot}{p})\) denotes the Legendre symbol and \(E_0, E_1, E_2, \ldots\) are the Euler numbers.

MSC:

11A07	Congruences; primitive roots; residue systems
11B65	Binomial coefficients; factorials; \(q\)-identities
11B68	Bernoulli and Euler numbers and polynomials

Keywords:

supercongruences; Apéry numbers and Apéry polynomials; generalized Domb numbers; Bernoulli numbers and Bernoulli polynomials; Euler numbers and Euler polynomials

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