On two congruences involving Franel numbers

Abstract

Via symbolic summation method, we establish the following series for ${\pi }^2$: \begin{align*} \sum_{k=1}^\infty \frac{H_k-2H_{2k}}{(-3)^k k} = \frac{π^2}{18}, \end{align*} 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.

12 pages

References in corpus (3)

• Congruences on sums of $q$-binomial coefficients
• Some new $q$-congruences for truncated basic hypergeometric series: even powers
• Infinite Summation Formulas Involving Riemann-Zeta function