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A short proof of a symmetry identity for the $(q,μ,ν)$-deformed Binomial distribution

We give a short and elementary proof of a $(q, μ, ν)$-deformed Binomial distribution identity arising in the study of the $(q, μ, ν)$-Boson process and the $(q, μ, ν)$-TASEP. This identity found by Corwin in [4] was a key technical step to prove an intertwining relation between the Markov transition matrices of these two classes of discrete-time Markov chains. This was used in turn to derive exact formulas for a large class of observables of both these processes.