Abstract
Let \(n\ge 3\) be an integer and p be a prime with \(p\equiv 1\pmod {n}\). In this paper, we show that
where the truncated hypergeometric series
and \(\Gamma _p\) denotes the p-adic Gamma function. This confirms a conjecture of Deines et al. (Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions, Directions in Number Theory, vol. 3, pp. 125–159. Assoc.WomenMath. Ser., Springer, New York, 2016). Furthermore, under the same assumptions, we also prove that
which solves another conjecture in [5].
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We are grateful to the anonymous referee for his/her very helpful comments on our paper.
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Chen Wang is supported by the National Natural Science Foundation of China (Grant No. 11971222). Hao Pan is supported by the National Natural Science Foundation of China (Grant No. 12071208). Hao Pan is the corresponding author.
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Wang, C., Pan, H. Supercongruences concerning truncated hypergeometric series. Math. Z. 300, 161–177 (2022). https://doi.org/10.1007/s00209-021-02772-0
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DOI: https://doi.org/10.1007/s00209-021-02772-0