Supercongruences for sums involving \binom ak^m

2022/04/21 by Zhi-Hong Sun, Sun, Zhi-Hong
Mathematics · #11B65 #11B68 #11E25 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11A07 #Secondary 05A19

paper · pdf · doi:10.48550/arxiv.2204.10132

openalex publication_date 2022/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime, and let a be a rational p-adic integer with a\not≡ 0\pmod p. In this paper, using WZ method we establish the congruences for ∑_k=0^p-1 \binom ak²(-1)^k(1-\frac 2ak) modulo p² and ∑_k=0^p-1 \binom ak^r(1-\frac 2ak)^s modulo p⁴, where r∈\3,4\ and s∈\1,3\.

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