Supercongruences involving products of two binomial coefficients

In this paper we deduce some new supercongruences modulo powers of a prime p>3p>3. Let d∈{0,1,…,(p−1)/2}d∈{0,1,…,(p−1)/2}. We show that∑k=0(p−1)/2(2kk)(2kk+d)8k≡0(modp)ifd≡p+12(mod2), and∑k=0(p−1)/2(2kk)(2kk+d)16k≡(−1p)+p2(−1)d4Ep−3(d+12)(modp3), where Ep−3(x)Ep−3(x) denotes the Euler polynomial of degree p−3p−3, and (−)(−) stands for the Legendre symbol. The paper also contains some other results such as∑k=0p−1k(1+(−1p))/2(6k3k)(3kk)864k≡0(modp2).