NEW SUPERCONGRUENCES INVOLVING PRODUCTS OF TWO BINOMIAL COEFFICIENTS

Abstract

Let $p>3$ be a prime and let $a$ be a rational $p$-adic integer with $a\not \equiv 0\;(\text{mod}\;p)$. We evaluate $$\begin{eqnarray}\mathop{\sum }_{k=1}^{(p-1)/2}\frac{1}{k}\binom{a}{k}\binom{-1-a}{k}\quad \text{and}\quad \mathop{\sum }_{k=0}^{(p-1)/2}\frac{1}{2k-1}\binom{a}{k}\binom{-1-a}{k}\end{eqnarray}$$ modulo $p^{2}$ in terms of Bernoulli and Euler polynomials.