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

TL;DR

This paper provides bijective proofs for two multivariable identities, Gould-Mohanty's and Raney-Mohanty's, using a word-based model to establish combinatorial equivalences.

Contribution

It introduces a novel bijective approach using word models to prove complex multivariable identities, extending classical combinatorial proofs.

Findings

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

Extension of classical identities to multivariable cases.

Use of word models for combinatorial proof techniques.

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 $$ \sum_{k=0}^{n}\frac{x}{x-kz}{x-kz\choose k}{y+kz\choose n-k}= {x+y\choose n}. $$