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

## Abstract

We prove some supercongruences involving the Apéry polynomials

$A_n (x) = \mathop {\mathop \sum \nolimits^ }\limits_n^{k = 0} \left( {{n \over k}} \right)^{2} \left( {{{n + k} \over k}} \right)^{2} x^k (n \in {\mathbb{N}} = \{ 0,1, \ldots ,\} ),$

the generalized Domb numbers

$D_n (A,B,C) = \mathop {\mathop \sum \nolimits^ }\limits_n^{k = 0} \left( {{n \over k}} \right)^{A} \left( {{{2k} \over k}} \right)^{B} \left( {{{2n - 2k} \over {n - k}}} \right)^{C} (n \in {\mathbb{N}})\quad {\rm{and}}\quad Q_n = \mathop {\mathop \sum \nolimits^ }\limits_n^{k = 0} \left( {{n \over k}} \right)\left( {{{n - k} \over k}} \right)\left( {{{n + k} \over k}} \right)(n \in {\mathbb{N}}),$

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

${{A_{p^r } ( - 1) - A_{p^{r - 1} } ( - 1)} \over {p^{3r} }} \equiv {{29} \over 6}B_{p - 3} \;(\bmod \;p)\quad {\rm{and}}\quad {{Q_{p^r } - Q_{p^{r - 1} } } \over {p^{3r} }} \equiv - {1 \over 9}B_{p - 3} \;(\bmod \;p),$

where $B_0 ,B_1 ,B_2 , \ldots$ are the Bernoulli numbers. The following supercongruences hold modulo $p$:

${{D_{p^r } (A,1,1) - D_{p^{r - 1} } (A,1,1)} \over {p^{(A + 1)r} }} \equiv \{ \matrix{ {8({{ - 1} \over {p^r }})E_{p - 3} ,\quad } & {{\rm{if}}A = 1,} \cr {} & {} \cr {{{16} \over 3}B_{p - 3} ,\quad } & {{\rm{if}}A = 2,} \cr }$

where $({ \cdot \over p})$ denotes the Legendre symbol and $E_0 ,E_1 ,E_2 , \ldots$ 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

Received: 17 June 2021; Revised: 2 September 2021; Accepted: 12 September 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441117
zbMATH: 1497.11013
Digital Object Identifier: 10.1216/rmj.2022.52.1105

Subjects:
Primary: 11A07 , 11B68 , 11E25
Secondary: 05A10 , 11B65 , 11B75

Keywords: Apéry numbers and Apéry polynomials , Bernoulli numbers and Bernoulli polynomials , Euler numbers and Euler polynomials , generalized Domb numbers , supercongruences