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Expressing symmetric function in power-sum basis

I am trying to prove the following identity \begin{equation} \prod_{i=1}^{m}(1-x_{i}z)^{-u}\prod_{j=1}^{n}(1-y_{i}z)^{-v} \prod_{i=1}^{m}\prod_{j=1}^{n}(1-(x_i +y_j)z)^{-w}\\ = \sum_{\lambda, \mu}c_{\...
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