Proof of two supercongruences conjectured by Z.-W.Sun involving Catalan-Larcombe-French numbers

Abstract

The harmonic numbers $H_n=\sum_{0 3$, 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 $p$, we have $$\sum_{k=0}^{p-1}\frac{P_k}{8^k}\equiv1+2\big(\frac{-1}p\big)p^2E_{p-3}\pmod{p^3}$$ and $$\sum_{k=0}^{p-1}\frac{P_k}{{16}^k}\equiv\big(\frac{-1}p\big)-p^2E_{p-3}\pmod{p^3},$$ where $P_n=\sum_{k=0}^n\frac{\binom{2k}k^2\binom{2(n-k)}{n-k}^2}{\binom nk}$ is the n-th Catalan-Larcombe-French number.