New super congruences involving Bernoulli and Euler polynomials

Abstract

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer with $a\not\equiv 0(\mathrm{mod}p)$. In this paper we establish congruences for $$\sum _{k=1}^{(p-1)/2}\frac{\left(\genfrac{}{}{0ex}{}{a}{k}\right)\left(\genfrac{}{}{0ex}{}{-1-a}{k}\right)}{k},\sum _{k=0}^{(p-1)/2}k\left(\genfrac{}{}{0ex}{}{a}{k}\right)\left(\genfrac{}{}{0ex}{}{-1-a}{k}\right)\text{and}\sum _{k=0}^{(p-1)/2}\frac{\left(\genfrac{}{}{0ex}{}{a}{k}\right)\left(\genfrac{}{}{0ex}{}{-1-a}{k}\right)}{2k-1}(\mathrm{mod}{p}^2)$$ in terms of Bernoulli and Euler polynomials. We also give some transformation formulas for congruences modulo $p^2$.

20 pages