Supercongruences involving Euler polynomials

Let p > 3 p>3 be a prime, and let a a be a rational p p -adic integer. Let { E n ( x ) } \{E_n(x)\} denote the Euler polynomials given by 2 e x t e t + 1 = ∑ n = 0 ∞ E n ( x ) t n n ! \frac {2\text {e}^{xt}}{\text {e}^t+1}=\sum _{n=0}^{\infty }E_n(x)\frac {t^n}{n!} . In this paper we show that a m p ; ∑ k = 0 p − 1 ( a k ) ( − 1 − a k ) ≡ ( − 1 ) ⟨ a ⟩ p + ( a − ⟨ a ⟩ p ) ( p + a − ⟨ a ⟩ p ) E p − 3 ( − a ) ( mod p 3 ) , a m p ; ∑ k = 0 p − 1 ( a k ) ( − 2 ) k ≡ ( − 1 ) ⟨ a ⟩ p − ( a − ⟨ a ⟩ p ) E p − 2 ( − a ) ( mod p 2 ) for a ≢ 0 ( mod p ) , \begin{align*} &\sum _{k=0}^{p-1}\binom ak\binom {-1-a}k\equiv (-1)^{\langle a\rangle _p}+ (a-\langle a\rangle _p)(p+a-\langle a\rangle _p)E_{p-3}(-a)\pmod {p^3}, \\&\sum _{k=0}^{p-1}\binom ak(-2)^k\equiv (-1)^{\langle a\rangle _p}-(a-\langle a\rangle _p)E_{p-2}(-a) \pmod {p^2}\quad \text {for}\quad a\not \equiv 0\pmod p, \end{align*} where ⟨ a ⟩ p ∈ { 0 , 1 , … , p − 1 } \langle a\rangle _p\in \{0,1,\ldots ,p-1\} satisfying a ≡ ⟨ a ⟩ p ( mod p ) a\equiv \langle a\rangle _p\pmod p . Taking a = − 1 3 , − 1 4 , − 1 6 a=-\frac 13,-\frac 14,-\frac 16 in the first congruence, we solve some conjectures of Z. W. Sun. We also establish a congruence for ∑ k = 0 p − 1 k ( a k ) ( − 1 − a k ) \sum _{k=0}^{p-1}k\binom ak\binom {-1-a}k modulo p 3 p^3 .