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NEW SUPERCONGRUENCES INVOLVING PRODUCTS OF TWO BINOMIAL COEFFICIENTS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 101 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 23 August 2019, pp. 367-378
- Print publication:
- June 2020
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- Article
- Export citation
-
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 modulo
$$\begin{eqnarray}\mathop{\sum }_{k=1}^{(p-1)/2}\frac{1}{k}\binom{a}{k}\binom{-1-a}{k}\quad \text{and}\quad \mathop{\sum }_{k=0}^{(p-1)/2}\frac{1}{2k-1}\binom{a}{k}\binom{-1-a}{k}\end{eqnarray}$$
$p^{2}$ in terms of Bernoulli and Euler polynomials.
