Proof of a supercongruence via the Wilf–Zeilberger method. (November 2021)

• Record Type: Journal Article
• Authors: Mao, Guo-Shuai
• Is Part Of: Journal of symbolic computation. Volume 107(2021)
• Page Start: 269
• Page End: 278
• Publication Date: 2021-11
• Subjects: 11A07 -- 11B68 -- 11B65 -- 33F10; Supercongruence -- Binomial coefficients -- Wilf–Zeilberger method -- Euler polynomials; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2021.04.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper, we prove a supercongruence via the Wilf–Zeilberger method and symbolic summation algorithms in the setting of difference rings. That is, for any prime p > 3, ∑ n = 0 ( p − 1 ) / 2 3 n + 1 ( − 8 ) n ( 2 n n ) 3 ≡ p ( − 1 p ) + p 3 4 ( 2 p ) E p − 3 ( 1 4 ) ( mod p 4 ), where ( ⋅ p ) stands for the Legendre symbol, and E n ( x ) are the Euler polynomials. This confirms a special case of a recent conjecture of Z.-W. Sun (Sun, 2019, (2.18)).