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On two congruences involving Franel numbers

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

Abstract

Via symbolic summation method, we establish the following series for \(\pi ^2\):

$$\begin{aligned} \sum _{k=1}^{\infty }\frac{H_k-2H_{2k}}{(-3)^k k}=\frac{\pi ^2}{18}, \end{aligned}$$

where \(H_k=\sum _{j=1}^k 1/j\). We also derive a p-adic congruence related to this series. As an application, we prove two congruences involving Franel numbers, one of which was originally conjectured by Sun.

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Acknowledgements

The author would like to thank the anonymous referee for his/her critical comments which helped to improve the exposition of the paper. This work was supported by the National Natural Science Foundation of China (grant 11801417).

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  1. Department of Mathematics, Wenzhou University, Wenzhou, 325035, People’s Republic of China

    Ji-Cai Liu

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Liu, JC. On two congruences involving Franel numbers. RACSAM 114, 201 (2020). https://doi.org/10.1007/s13398-020-00935-y

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