Non integer derivative of $1/p(x)$ I need to find the $k$'th derivative of $1/p(x)$, where $p(x)$ is a polynomial and $k\in\mathbb{R}$
It dosen't have to be an explicit formula, an algorithm which finds a formula for some $k$ is fine.

Let me sketch an idea for you, based on Fourier transforms: let $f$ be a function rapidly decreasing to infinity and let $\mathcal{F}(f)(y)=\frac{1}{\sqrt{2 \pi}} \int\mathbb{e}^{\mathbb{i}yx}f(x)$ denote its Fourier transform. Notice that $\mathcal{F}(f^{(n)})(y)=(-\mathbb{i})^n y^n \mathcal{F}(f)(y)$. Then $f^{(n)}=(-\mathbb{i})^n  \mathcal{F}^{-1} (y^n \mathcal{F}(f))$. This means that, for those $\alpha \in \mathbb{R}$ for which the formula meakes sense, you can define $f^{(\alpha)}=(-\mathbb{i})^\alpha  \mathcal{F}^{-1} (y^\alpha \mathcal{F}(f))$, where $(-\mathbb{i})^\alpha=\mathbb{e}^{\frac{3 \pi}{2} \mathbb{i} \alpha}$. Attention must be payed at the roots of $p$! Also, if you don't like the presence of $\mathbb{i}$, you can use Laplace's transform $\mathcal{L}(f)(y)=\int\mathbb{e}^{-yx}f(x)$, but then you'll have to restrict $p$ to just $(-\infty,0)$ or $(0, \infty)$.
Note that, unlike in the answers based on the Gammma function, $\alpha$ is not restricted to be in $(0,1)$.
