I am trying to prove this identity between rational functions involving symmetrization among variables. Let us consider a set of variables $\{p_1,\ldots,p_n\}$, which I indicate globally as $\mathbf{p}$. We define the function
$$ f(\mathbf{p}) = (p_1 - p_2)(p_1 - p_2 + k) \prod_{i<j} \frac{p_i -p_j}{p_i - p_j + k} $$
Then we consider the operator of symmetrization applied to this function $$ \mathcal{S} f (\mathbf{p}) = \frac{1}{n!}\sum_{\sigma} f(\sigma p) \;. $$
I would like to prove that $$ \mathcal{S} f (\mathbf{p}) = \frac{2}{n(n-1)}\prod_{i<j}\frac{(p_i -p_j)^2}{(p_i-p_j)^2-k^2} \sum_{i<j} (p_i - p_j)^2 - (i-j)^2 k^2 $$
I have the feeling that it can be proven by induction, but I was wondering if there is a smarter and faster way.