Proof of two supercongruences conjectured by Z.-W.Sun involving Catalan-Larcombe-French numbers
arXiv:1511.06222 · doi:10.1016/j.jnt.2017.03.017
Abstract
The harmonic numbers $H_n=\sum_{0<k\ls n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. With helps of some combinatorial identities, we establish the following two congruences: $$\sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_k}{(2k+1)16^k}\ \mbox{modulo}\ p^2\ \mbox{and}\ \sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_{2k}}{(2k+1)16^k}\ \mbox{modulo}\ p$$ for any prime , the second one was conjectured by Z.-W. Sun in 2012. These two congruences are very important to prove the following conjectures of Z.W.Sun: For any old prime , we have and where is the n-th Catalan-Larcombe-French number.
9 pages, 0 figures