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In a paper I'm reading, I found the following formula for the coefficients of the $m$th order central difference scheme. For all $\ell \in \{-m,-m+1, \dots, m-1,m\}$: $$a_{\ell} = \begin{cases} \frac{(-1)^{\ell-1}}{\ell} \cdot \frac{\binom{m}{|\ell|}}{\binom{m+|\ell|}{|\ell|}}, & \text{if } \ell \neq 0 \\ 0, & \text{otherwise} \end{cases}$$ In the paper, it is not proven that these are indeed the coefficients of the $m$th order central difference scheme, so I want to prove this myself, i.e., I want to prove for $k \in \{0,1, \dots, 2m\}$: $$\sum_{\ell=-m}^m a_{\ell}\ell^k = \begin{cases} 1, & \text{if } k = 1 \\ 0, & \text{otherwise} \end{cases}$$ Numerical simulation strongly suggests that it's true, however I have no idea how one would go about proving it. Any ideas would be greatly appreciated.

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