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Divisibility of some binomial sums

Volume 194 / 2020

He-Xia Ni, Hao Pan Acta Arithmetica 194 (2020), 367-381 MSC: Primary 11B65; Secondary 05A10, 05A30, 11A07. DOI: 10.4064/aa181114-24-7 Published online: 30 March 2020

Abstract

With the help of $q$-congruences, we consider divisibility of some binomial sums. For example, for any integers $\rho \geq 2$ and $n\geq 2$, \begin{equation} \sum _{k=0}^{n-1}(4k+1)\binom {2k}{k}^\rho \cdot (-4)^{\rho (n-1-k)}\equiv 0\ \biggl (\!{\rm mod}\,{2^{\rho -2}n\binom {2n}{n}}\biggr ).\tag*{$(*)$} \end{equation} In fact, we obtain a general result concerning divisibility of $q$-binomial sums. Using this result, we also confirm a $q$-analogue of ($*$), which was conjectured by Guo.

Authors

  • He-Xia NiDepartment of Applied Mathematics
    Nanjing Audit University
    Nanjing 211815
    People’s Republic of China
    e-mail
  • Hao PanSchool of Applied Mathematics
    Nanjing University of Finance and Economics
    Nanjing 210023
    People’s Republic of China
    e-mail

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