Suppose $f$ is a smooth function.
$m$ is a even integer.
$2p$-order finite difference of $m$-order derivative of $f(x)$ can be approximated by $$D^mf(x) \approx \frac{1}{h^m}\sum_{n=-L}^{L}\alpha_{p,n}^m E^nf(x)$$
where $L = p+[(m-1)/2]$ ( [$\cdot$] represent the integer part.)
$Df(x)=f^{'}(x),Ef(x)=f(x+h)$.
I need to deduce the formula of $\alpha_{\infty,n}^m$, which means $\lim _{p\to \infty}\alpha_{p,n}^m $.
If your formula and its coefficients is valid for all smooth functions, then it has to be valid for band-limited functions where you have the reconstruction formula \begin{align} f(x+s)&=\sum_{k\in\Bbb Z} f(x+kh)\frac{\sin(\frac{\pi}{h}(s-kh))}{\frac{\pi}{h}(s-kh)} \\ &=\sum_{k\in\Bbb Z} (-1)^k f(x+kh)\frac{\sin(\frac{\pi s}{h})}{\frac{\pi}{h}(s-kh)} \\ &=f(x)\frac{h}{\pi s}\sin(\frac{\pi s}{h}) + \frac1\pi\sin(\frac{\pi s}{h})\sum_{0\ne k\in\Bbb Z} (-1)^{k+1} f(x+kh)\sum_{m=0}^\infty \frac{s^m}{h^mk^{m+1}} \end{align} Now one can compare coefficients of the powers of $s$ to find the expressions for the derivatives of $f$ in $x$. For instance, for the first derivative we need the terms of the geometric series with $m=0$, $$ f'(x)=-\frac{1}{h}\sum_{0\ne k\in\Bbb Z} \frac{(-1)^{k+1}}{k} f(x+kh) $$ For the second derivative $$ f''(x)=-\frac{\pi^2}{3h^2}+\frac{2}{h^2}\sum_{0\ne k\in\Bbb Z} \frac{(-1)^{k+1} }{k^2}f(x+kh) $$ etc.