Supercongruences for sums involving Domb numbers

Abstract

We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any prime p≥5,∑k=0p−13k+1(−32)kDomb(k)≡(−1)p−12p+p3Ep−3(modp4), which is regarded as a p-adic analogue of the interesting formula for 1/π due to Rogers:∑k=0∞3k+1(−32)kDomb(k)=2π. Here Domb(n) and En are the famous Domb numbers and Euler numbers.