Factors of alternating sums of powers of q-Narayana numbers

JOURNAL OF NUMBER THEORY

卷:177

Article

Guo, Victor J. W.¹  Jiang, Qiang-Qiang² 

[1] Huaiyin Normal Univ, Sch Math Sci, Huaian 223300, Jiangsu, Peoples R China

[2] East China Normal Univ, Dept Math, Shanghai 200241, Peoples R China

关键词: q-binomial coefficients;    q-Narayana numbers;    q-Catalan numbers;   

DOI  :  10.1016/j.jnt.2017.01.009

来源: Elsevier

【 摘 要 】

The q-Narayana numbers N-q (n, k) and q-Catalan numbers C-n(q) are respectively defined by N-q (n, k) =1 - q/1 - q(n) [(n)(k)] [(n)(k - 1)] and C-n(q) = 1 - q / 1 - q(n+1) [(2n)(n)], where [(n)(k)] = Pi(k)(i=1) 1-q(n-i+1) / 1-q(i) .We prove that, for any positive integers n and r, there holds Sigma(n)(k=-n) (-1)(k)q(jk2+(2k)) N-q(2n + 1, n + k + 1)(r) equivalent to 0 (mod C-n(q)), where 0 <= j <= 2r - 1. We also propose several related conjectures. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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