Translator Disclaimer
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

An(x)=k=0nnk2n+kk2xk(n={0,1,,}),

the generalized Domb numbers

Dn(A,B,C)=k=0nnkA2kkB2n2knkC(n) andQn=k=0nnknkkn+kk(n),

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

Apr(1)Apr1(1)p3r296Bp3(modp) andQprQpr1p3r19Bp3(modp),

where B0,B1,B2, are the Bernoulli numbers. The following supercongruences hold modulo p:

Dpr(A,1,1)Dpr1(A,1,1)p(A+1)r{8(1pr)Ep3, if A=1,163Bp3, if A=2,

where (p) denotes the Legendre symbol and E0,E1,E2, are the Euler numbers.

Citation

Download 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

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

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

JOURNAL ARTICLE
22 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.52 • No. 3 • June 2022
Back to Top