Super congruences involving Bernoulli and Euler polynomials

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer. Let $\{B_n(x)\}$ and $\{E_n(x)\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\sum _{k=0}^{p-1}\left(\genfrac{}{}{0px}{}{a}{k}\right)\left(\genfrac{}{}{0px}{}{-1-a}{k}\right)\equiv (-1{)}^{⟨a{⟩}_p}+p^{2}t(t+1)E_{p-3}(-a)(\mathrm{m}\mathrm{o}\mathrm{d}{p}^3)$$ and for $a\equiv ̸-\frac{1}{2}(\mathrm{m}\mathrm{o}\mathrm{d}p)$, $$\sum _{k=0}^{p-1}\left(\genfrac{}{}{0px}{}{a}{k}\right)\left(\genfrac{}{}{0px}{}{-1-a}{k}\right)\frac{1}{2k+1}\equiv \frac{1+2t}{1+2a}+p^2\frac{t(t+1)}{1+2a}{B}_{p-2}(-a)(\mathrm{m}\mathrm{o}\mathrm{d}{p}^3),$$ where $\langle a\rangle_p\in{0,1,\ldots,p-1}$satisfying$a\equiv \langle a\rangle_p\pmod p$and$t=(a-\langle a\rangle_p)/p$\mathrm{.}Taking$a=-\frac 13,-\frac 14,-\frac 16$ in the above congruences we solve some conjectures of Z.W. Sun. In this paper we also establish congruences for $\sum_{k=0}^{p-1}k\binom ak\binom{-1-a}k,
\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\frac 1{2k-1},\ \sum_{k=1}^{p-1}\frac 1k\binom ak\binom{-1-a}k\pmod{p^3}$and$\sum_{k=1}^{p-1}\frac {(-1)^k}k\binom ak,\ \sum_{k=0}^{p-1}\binom ak(-2)^k\pmod{p^2}.$