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Proof of a conjecture of Sun and its extension by Guo

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Abstract

In this paper, we mainly prove the following result: For any positive integers l and n and nonnegative integer k with \(k\le n-1\), we have

$$\begin{aligned} (2l-1)!!\sum _{m=k}^{n-1}(2m+1)^{2l-1}\left( {\begin{array}{c}m+k\\ 2k\end{array}}\right) \left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2\equiv 0 \left( \textrm{mod} n^2\left( {\begin{array}{c}n-1\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) \right) . \end{aligned}$$

This confirms a conjecture of Sun and its extension by Guo.

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Acknowledgements

The author is grateful to the anonymous referee for their careful reading and valuable comments.

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  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Wei Xia

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  1. Wei Xia
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Correspondence to Wei Xia.

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Supported by the National Natural Science Foundation of China (Grant No. 11971222).

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Xia, W. Proof of a conjecture of Sun and its extension by Guo. Ramanujan J 62, 617–631 (2023). https://doi.org/10.1007/s11139-022-00668-z

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