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ON THE DIVISIBILITY OF SUMS INVOLVING APÉRY-LIKE POLYNOMIALS

Part of: Elementary number theory Combinatorics

Published online by Cambridge University Press:  14 March 2022

SHENG YANG  and
JI-CAI LIU Open the ORCID record for JI-CAI LIU [Opens in a new window]
SHENG YANG
Affiliation:
Department of Mathematics, Wenzhou University, Wenzhou 325035, PR China e-mail: ysdj@sina.com
JI-CAI LIU*
Affiliation:
Department of Mathematics, Wenzhou University, Wenzhou 325035, PR China
*
e-mail: jcliu2016@gmail.com
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Abstract

We prove a divisibility result on sums involving the Apéry-like polynomials

$$ \begin{align*} V_n(x)=\sum_{k=0}^n {n\choose k}{n+k\choose k}{x\choose k}{x+k\choose k}, \end{align*} $$

which confirms a conjectural congruence of Z.-H. Sun. Our proof relies on some combinatorial identities and transformation formulae.

Keywords

congruencesApéry numbersChu–Vandermonde identity

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 05A19: Combinatorial identities, bijective combinatorics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 203 - 208
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The second author was supported by the National Natural Science Foundation of China (grant 12171370).

References

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