I have a general function, U, which has the following form, $$ U(r-r^{'}) $$ where r and r' are defined as, $$ r = R + u(R) \\ r^{'} = R^{'} + u(R^{'}) $$ For every R and R', u is just small variations around them and it is a function of 3 coordinate variables, x, y, and z.
What is the Taylor expansion of U? Also, I need it as a compact summation expression.
In general, if you have a function $f: \mathbb{R}^d\to \mathbb{R}$ (which is how the potential energy works, from a vector to a scalar), then $$ f(\vec{x}+\vec{h})=\sum_{n=0}^\infty \frac{1}{n!}(\vec{h}\cdot \vec{\nabla})^nf(\vec{x}) $$ In your problem, $d=3$ i.e. 3D space, $f=U$ potential energy, $\vec{x}=\vec{R}-\vec{R}'$ and $\vec{h}=\vec{u}(\vec{R})-\vec{u}(\vec{R}')$.