2017/05/08 by Victor J. W. Guo, Guo, Victor J. W., Su-Dan Wang +1
Mathematics · #05A30 #11B65 #65Q05 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1705.06236
openalex publication_date 2017/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that, for all positive integers n1, …, nm, nm+1=n1, and non-negative integers j and r with j\leqslant m, the following two expressions amp;(1)/([n1+nm+1])n1+nm\brack n1-1∑k=0n1 qj(k2+k)-(2r+1)k[2k+1]2r+1∏i=1m ni+ni+1+1\brack ni-k,
amp;(1)/([n1+nm+1])n1+nm\brack n1-1∑k=0n1(-1)k q^k\choose 2+j(k2+k)-2rk[2k+1]2r+1∏i=1m ni+ni+1+1\brack ni-k are Laurent polynomials in q with integer coefficients, where [n]=1+q+⋯+qn-1 and n\brack k=∏i=1k(1-qn-i+1)/(1-qi). This gives a q-analogue of some divisibility results of sums and alternating sums involving binomial coefficients and powers of integers obtained by Guo and Zeng. We also confirm some related conjectures of Guo and Zeng by establishing their q-analogues. Several conjectural congruences for sums involving products of q-ballot numbers (2n\brack n-k-2n\brack n-k-1) are proposed in the last section of this paper.