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  • ZHI-HONG SUN (a1)


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.



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The author is supported by the National Natural Science Foundation of China (grant no. 11771173).



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