Bijective proofs of Gould-Mohanty's and Raney-Mohanty's identities

Abstract

Using the model of words, we give bijective proofs of Gould-Mohanty's and Raney-Mohanty's identities, which are respectively multivariable generalizations of Gould's identity $$\sum_{k=0}^{n}{x-kz\choose k}{y+kz\choose n-k}= \sum_{k=0}^{n}{x+\epsilon-kz\choose k}{y-\epsilon+kz\choose n-k} $$ and Rothe's identity $$ \sum_{k=0}^{n}\frac{x}{x-kz}{x-kz\choose k}{y+kz\choose n-k}= {x+y\choose n}. $$