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I am stuck trying to prove the following statement:

Let $N,J \in \mathbb{N}\cup\{0\}$ with $N \geq J$ and let $k_1,...,k_N\in \mathbb{R}$. I define ($i$ is the imaginary unit) $$ R_J := \prod_{m=1}^J \frac{-i}{\sum_{j=1}^{m} k_{J+1-j}}, \qquad L_{N,J} := \prod_{m=J+1}^N \frac{i}{\sum_{j=J+1}^{m} k_j}, \qquad R_0 = L_{N,N} =1. $$ Then we have
$$ \sum_{J= 0}^N R_J L_{N,J} = 0 ~\text{for } N\geq 1. $$

The case $N=1$ is trivial. For the induction step this is how far I have got:
Suppose the statement is true for $N-1$. Then
$$ \sum_{J= 0}^N R_J L_{N,J} = \sum_{J= 0}^{N-1} R_J L_{N,J} + R_N = \sum_{J= 0}^{N-1} R_J L_{N-1,J} \Big{(}\frac{i}{k_{J+1} + ... + k_N} + C \Big{)} + R_N, $$ where $C$ can be any complex number that does not depend on $J$. Maybe one has to choose $C$ in some clever way to show that the whole expression gives zero...

I checked this statement for large values of $N$ in mathematica, so it should be correct. I think the problem should not be very hard, anyone see the solution?

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