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Supercongruences involving products of three binomial coefficients

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Let \(p>3\) be a prime, and let a be a rational p-adic integer. Using the WZ method we establish the congruences for \(\sum _{k=0}^{p-1} \left( {\begin{array}{c}a\\ k\end{array}}\right) \left( {\begin{array}{c}-1-a\\ k\end{array}}\right) \left( {\begin{array}{c}2k\\ k\end{array}}\right) \frac{w(k)}{4^k}\) modulo \(p^3\), where \(w(k)\in \{1,\frac{1}{k+1},\) \(\frac{1}{(k+1)^2},\frac{1}{2k-1}\}\). Taking \(a=-\frac{1}{2},-\frac{1}{3},-\frac{1}{4},-\frac{1}{6}\) in the congruences confirms some conjectures posed by the author earlier.

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Funding

The author was supported by the National Natural Science Foundation of China (Grant No. 12271200).

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  1. School of Mathematics and Statistics, Huaiyin Normal University, Huaian, 223300, Jiangsu, People’s Republic of China

    Zhi-Hong Sun

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Sun, ZH. Supercongruences involving products of three binomial coefficients. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 131 (2023). https://doi.org/10.1007/s13398-023-01458-y

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