On sums involving products of three binomial coefficients

Abstract

In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{2k}{k+d}}{64^k}\equiv 0\pmod{p^2}$$ for all $d\in\{0,\ldots,p-1\}$ with $d\equiv (p+1)/2\pmod2$. If $p\equiv 1\pmod4$ and $p=x^2+y^2$ with $x\equiv 1\pmod4$ and $y\equiv 0\pmod2$, then we show $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{2k}{k+1}}{(-8)^k}\equiv 2p-2x^2\pmod{p^2}\ \ \mbox{and}\ \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{2k}{k+1}^2}{(-8)^k}\equiv-2p\pmod{p^2}$$ by means of determining $x$ mod $p^2$ via $$(-1)^{(p-1)/4}\,x\equiv\sum_{k=0}^{(p-1)/2}\frac{k+1}{8^k}\binom{2k}k^2\equiv\sum_{k=0}^{(p-1)/2}\frac{2k+1}{(-16)^k}\binom{2k}k^2\pmod{p^2}.$$ We also solve the remaining open cases of Rodriguez-Villegas' conjectural congruences on $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{3k}k}{108^k},\ \ \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{256^k}, \ \ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}\binom{3k}k\binom{6k}{3k}}{12^{3k}}$$ modulo $p^2$.