There is a well-known delta function identity which allows for the expansion of
$$\chi_\epsilon(x)\equiv\left(\frac{1}{\epsilon^2+x^2}\right)^a,\quad x\in \mathbb{R}^n$$
see for example this Math.SE thread. I'm interested in the following generalization. First, we take $X_i\in \mathbb{R}^{n+1}$ as $X_i=(x_i,\xi_i)\in \mathbb{R}^n\times \mathbb{R}$. We then let $f:\mathbb{R}^{n+1}\times\cdots \times \mathbb{R}^{n+1}\to\mathbb{R}$ with $N$ factors of $\mathbb{R}^{n+1}$ be defined by
$$f(X_1,\dots, X_N)\equiv \prod_{i<j}^N\left(\frac{1}{|x_{ij}|^2+\xi_{ij}^2}\right)^{a_{ij}}$$
where $x_{ij}\equiv x_i-x_j$ and $\xi_{ij}=\xi_i-\xi_j$. I want to consider the expansion of $f(X_1,\dots, X_n)$ in powers of $\xi_{12}$ in the distributional sense.
The previous identity is recovered when $N=2$, where we can identify $\chi_\epsilon(x)=f((x,\epsilon),(0,0))$ and $a=a_{12}$.
Now the thing is that the first thought I had to expand this function as I suggested was to try to use the previous result and multiply expansions together. I guess this doesn't work because it is a multiplication of distributions, which is in general quite subtle.
As such, how can I write down the complete Taylor expansion of $f(X_1,\dots, X_N)$ in $\xi_{12}$?