Super congruences involving Bernoulli and Euler polynomials

Abstract

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 $$\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\equiv (-1)^{\langle a\rangle_p}+ p^2t(t+1)E_{p-3}(-a)\pmod{p^3}$$ and for $a\not\equiv -\frac 12\pmod p$, $$\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\frac 1{2k+1}\equiv \frac{1+2t}{1+2a} +p^2\frac{t(t+1)}{1+2a}B_{p-2}(-a)\pmod{p^3},$$ where $\langle a\rangle_p\in\{0,1,\ldots,p-1\}$ satisfying $a\equiv \langle a\rangle_p\pmod p$ and $t=(a-\langle a\rangle_p)/p$. Taking $a=-\frac 13,-\frac 14,-\frac 16$ in the above congruences we solve some conjectures of Z.W. Sun. We also prove that for $a\not\equiv 0,\pm \frac 12\pmod p$, $$\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\frac 1{2k-1}\equiv -(2a+1)(2t+1)-p^2t(t+1)(4+(2a+1)B_{p-2}(-a)) \pmod{p^3}. $$