  In this paper we deduce some new supercongruences modulo powers of a prime
$p>3$. Let $d\in\{0,1,\ldots,(p-1)/2\}$. We show that
  $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{2k}{k+d}}{8^k}\equiv 0\
(\mbox{mod}\ p)\ \ \ \mbox{if}\ d\equiv \frac{p+1}2\ (\mbox{mod}\ 2),$$ and
$$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{2k}{k+d}}{16^k}
\equiv\left(\frac{-1}p\right)+p^2\frac{(-1)^d}4E_{p-3}\left(d+\frac12\right)\pmod{p^3},$$
where $E_{p-3}(x)$ denotes the Euler polynomial of degree $p-3$, and $(-)$
stands for the Legendre symbol. The paper also contains some other results such
as
$$\sum_{k=0}^{p-1}k^{(1+(\frac{-1}p))/2}\frac{\binom{6k}{3k}\binom{3k}k}{864^k}\equiv0\pmod{p^2}.$$

Number Theory;Combinatorics