An identity for rational functions leading to equations for multiple polylogarithms

Question

The following identity is not hard to prove: $$ \sum_{1\leq i_1<i_2<\ldots <i_{2n}\leq N} (-1)^{i_1+\ldots+i_{2n}}\frac{(1-x_{i_1})(1-x_{i_3})\ldots(1-x_{i_{2n-1}})}{(1-x_{i_2})(1-x_{i_4}) \ldots (1-x_{i_{2n}})\ \ }= \frac{(x_{1}-x_2)(x_{2}-x_{3}) \ldots (x_{N-1}-x_{N})}{(1-x_{2})\ \ldots\ \ \ \ (1-x_{N-1})(1-x_{N})}. $$ I am curious if it appeared anywhere before and if it is a part of a certain family of similar identities. As a corollary, one immediately sees that LHS is a power series starting with degree $N-1.$ This fact leads to a family of functional equations for multiple polylogarithms.

Answer 1

If we denote $1-x_i=y_i$, this reads as $$\sum_{1\leq i_1<i_2<\ldots <i_{2n}\leq N} (-1)^{i_1+\ldots+i_{2n}}\frac{y_{i_1}y_{i_3}\ldots y_{i_{2n-1}}}{y_{i_2}y_{i_4} \ldots y_{i_{2n}}}= \frac{(y_{2}-y_1)(y_{3}-y_{2}) \ldots (y_{N}-y_{N-1})}{y_{2}\ldots y_{N-1}y_{N}}\\ =(1-y_1/y_2)(1-y_2/y_3)\ldots (1-y_{N-1}/y_N),$$ which may be proved by expanding the brackets. And such expanding was certainly done before. I am less sure about $1-y_i=x_i$ change of variables.

The coefficients of products of differences often appear, perphaps the most known is Vandermonde determinant $$\prod_{0\leqslant i<j\leqslant n-1}(y_j-y_i)=\sum_{\pi}{\rm sign}\,(\pi)\prod y_i^{\pi_i},$$ where $\pi$ runs over all $n!$ permutations of the set $\{0,\ldots,n-1\}$. There are many interesting cases, when all coefficients can not be found, but some of them can, like Dyson's conjecture (I have no idea whether it can lead to anything interesting about polylogarithms.) If you need complete expansions only, there are not so many cases when they are known, apart of above I can suggest the cycle graph instead of the path. If you allow not only differences, there are complete expansions of, for example, $(\sum x_i)^N \prod_{i<j}(x_j-x_i)$.