Bulletin of the
Korean Mathematical Society BKMS

In this paper, we give new congruences with the generalized Catalan numbers and harmonic numbers modulo $p^{2}.$ One of our results is as follows: for prime number $p>3,$ \begin{align*} &\sum\limits_{k=(p+1)/2}^{p-1}k^{2}B_{p,k}B_{p,k-(p-1)/2}H_{k}\equiv \left( -1\right) ^{(p-1)/2}\left( -\frac{521}{36}p-\frac{1}{p}-\frac{41}{12}\right. \\ &\text{ \ \ \ \ \ \ \ }\left. +pH_{3(p-1)/2}^{2}-10pq_{p}^{2}(2)+4\left( \frac{10}{3}p+1\right) q_{p}(2)\right) \pmod{p^{2}}, \end{align*} where $q_{p}(2)$ is Fermat quotient.

Keywords: Congruences, harmonic numbers and binomial coefficients

MSC numbers: Primary 11B50, 11A07, 11B65