On the supercongruences involving harmonic numbers of order 2

2022/01/10 by Guo-Shuai Mao, Hao Pan, Mao, Guo-Shuai +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2201.03418

openalex publication_date 2022/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove several supercongruences involving the harmonic number of order two H_n⁽²⁾:=∑_k=1ⁿ1/k². For example, if p>5 is prime and α is p-integral, then we can completely determine ∑_k=0^p-1\fracH_k⁽²⁾k⋅\binomαk\binom-1-αk\quadand ∑_k=0^(p-1)/(2)\fracH_k⁽²⁾k⋅\binomαk\binom-1-αk modulo p³. In particular, by setting α=-1/2, we confirm two conjectured congruences of Z.-W. Sun.

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