combinatorial identities – An identification for rational features resulting in equations for a number of polylogarithms retort

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combinatorial identities – An identification for rational features resulting in equations for a number of polylogarithms

The following identification isn’t troublesome to show:
$$
sum_{1leq i_1<i_2<ldots <i_{2n}leq N} (-1)^{i_1+ldots+i_{2n}}frac{(1-x_{i_1})(1-x_{i_3})ldots(1-x_{i_{2n-1}})}{(1-x_{i_2})(1-x_{i_4})ldots (1-x_{i_{2n}}) }=
frac{x_{1}(x_{2}-x_{3})(x_{3}-x_{4})ldots (x_{N-1}-x_{N})}{(1-x_{1})(1-x_{2})ldots (1-x_{N-1})(1-x_{N})}.
$$

I’m inquisitive if it appeared anyplace earlier than and if it’s a sever of a inescapable household of comparable identities. As a corollary, one instantly sees that LHS is an influence succession beginning with diploma $N-1.$ This truth results in a household of purposeful equations for a number of polylogarithms.

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