Congruences involving generalized Catalan numbers and Bernoulli numbers

Abstract

<abstract><p>In this paper, we establish some congruences mod $ p^3 $ involving the sums $ \sum_{k = 1}^{p-1}k^mB_{p, k}^{2l} $, where $ p > 3 $ is a prime number and $ B_{p, k} $ are generalized Catalan numbers. We also establish some congruences mod $ p^2 $ involving the sums $ \sum_{k = 1}^{p-1}k^mB_{p, k}^{2l_1}B_{p, k-d}^{2l_2} $, where $ m, l_1, l_2, d $ are positive integers and $ 1\leq d\leq p-1 $.</p></abstract>