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We give a short and elementary proof of a $(q, \mu, \nu)$-deformed Binomialdistribution identity arising in the study of the $(q, \mu, \nu)$-Boson processand the $(q, \mu, \nu)$-TASEP. This identity found by Corwin in [4] was a keytechnical step to prove an intertwining relation between the Markov transitionmatrices of these two classes of discrete-time Markov chains. This was used inturn to derive exact formulas for a large class of observables of both theseprocesses.

A short proof of a symmetry identity for the $(q,\mu,\nu)$-deformed Binomial distribution