Affiliation:
1. Department of Mathematics, Luoyang Normal College, Luoyang 471934, China
2. Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China
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>
Publisher
American Institute of Mathematical Sciences (AIMS)
Reference21 articles.
1. E. Deutsch, L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241–265. http://dx.doi.org/10.1016/S0012-365X(01)00121-2
2. L. Elkhiri, S. Koparal, N. Ömür, New congruences with the generalized Catalan numbers and harmonic numbers, Bull. Korean Math. Soc., 58 (2021), 1079–1095. http://dx.doi.org/10.4134/BKMS.b200359
3. J. W. L. Glaisher, On the residues of the sums of products of the first $p-1$ numbers and their powers, to modulus $p^2$ or $p^3$, Quarterly J. Math., 31 (1900), 321–353.
4. H. W. Gould, Combinatorial Identity, New York: Morgantown Printing and Binding Co., 1972.
5. J. W. Guo, J. Zeng, Factors of binomial sums from the Catalan triangle, J. Number Theory, 130 (2010), 172–186. http://dx.doi.org/10.1016/j.jnt.2009.07.005