New congruences for sums involving Apery numbers or central Delannoy numbers

2010/08/17 by Victor J. W. Guo, Jiang Zeng, Guo, Victor J. W. +1 · 1 citation
Mathematics · #05A10 #11A07 #11B65 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1008.2894

openalex publication_date 2010/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Apéry numbers A_n and central Delannoy numbers D_n are defined by A_n=∑_k=0ⁿn+k\choose 2k²2k\choose k², D_n=∑_k=0ⁿn+k\choose 2k2k\choose k. Motivated by some recent work of Z.-W. Sun, we prove the following congruences: ∑_k=0^n-1(2k+1)^2r+1A_k &≡ ∑_k=0^n-1ε^k (2k+1)^2r+1D_k ≡ 0\pmod n, where n\geqslant 1, r\geqslant 0, and ε=±1. For r=1, we further show that ∑_k=0^n-1(2k+1)³A_k &≡ 0\pmodn³, ∑_k=0^p-1(2k+1)³A_k &≡ p³ \pmod2p⁶, where p>3 is a prime. The following congruence ∑_k=0^n-1 n+k\choose k²n-1\choose k² ≡ 0 \pmodn plays an important role in our proof.

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