Proof of a conjecture involving Sun polynomials

2015/10/27 by Victor J. W. Guo, Guo, Victor J. W., Guo-Shuai Mao +3
Mathematics · #05A10 #11A07 #11B65 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A10 #msc:11A07 #msc:11B65

paper · pdf · doi:10.48550/arxiv.1511.04005

15 pages

openalex publication_date 2015/10/27 · arxiv created 2015/12/28 · arxiv updated 2015/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Sun polynomials g_n(x) are defined by g_n(x)=∑_k=0ⁿn\choose k²2k\choose kx^k. We prove that, for any positive integer n, there hold (1)/(n)∑_k=0^n-1(4k+3)g_k(x) ∈ℤ[x],\quadand
∑_k=0^n-1(8k²+12k+5)g_k(-1)≡ 0\pmodn. The first one confirms a recent conjecture of Z.-W. Sun, while the second one partially answers another conjecture of Z.-W. Sun. We give three different proofs of the former. One of them depends on the following congruence: m+n-2\choose m-1n\choose m2n\choose n≡ 0\pmodm+n\quadfor m,n\geqslant 1.

Related

• Proof of a recent conjecture of Z.-W. Sun
• Proof of some conjectures of Z.-W. Sun on the divisibility of certain double-sums
• q-Congruences for Z.-W. Sun's generalized polynomials w^(α)_k(x)
• Proof of two congruence conjectures of Z.-W. Sun
• On a conjectural supercongruence involving the dual sequence s_n(x)