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Symbolic summation methods and hypergeometric supercongruences
Journal of Mathematical Analysis and Applications ( IF 1.2 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.jmaa.2020.124068
Chen Wang

Abstract In this paper, via the WZ method and the summation package Sigma , we establish the following two supercongruences: ∑ k = 0 ( p + 1 ) / 2 ( 3 k − 1 ) ( − 1 2 ) k 2 ( 1 2 ) k 4 k k ! 3 ≡ p − 6 p 3 ( − 1 p ) + 2 p 3 ( − 1 p ) E p − 3 ( mod p 4 ) , ∑ k = 0 p − 1 ( 3 k − 1 ) ( − 1 2 ) k 2 ( 1 2 ) k 4 k k ! 3 ≡ p − 2 p 3 ( mod p 4 ) , where p > 3 is a prime, E p − 3 is the ( p − 3 ) -th Euler number and ( − 1 p ) = ( − 1 ) ( p − 1 ) / 2 is the Legendre symbol. The first congruence modulo p 3 confirms a recent conjecture of Guo and Schlosser.

中文翻译:

符号求和方法和超几何超同余

摘要 在本文中,通过 WZ 方法和求和包 Sigma,我们建立了以下两个超同余: ∑ k = 0 ( p + 1 ) / 2 ( 3 k − 1 ) ( − 1 2 ) k 2 ( 1 2 ) k 4 kk !3 ≡ p − 6 p 3 ( − 1 p ) + 2 p 3 ( − 1 p ) E p − 3 ( mod p 4 ) , ∑ k = 0 p − 1 ( 3 k − 1 ) ( − 1 2 ) k 2 ( 1 2 ) k 4 kk !3 ≡ p − 2 p 3 ( mod p 4 ) ,其中 p > 3 是素数,E p − 3 是 ( p − 3 ) -th Euler 数并且 ( − 1 p ) = ( − 1 ) ( p − 1 ) / 2 是勒让德符号。第一个同余模 p 3 证实了郭和施洛瑟最近的猜想。
更新日期:2020-08-01
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