Bijective Proofs of Gould's and Rothe's Identities

TL;DR

This paper presents bijective proofs for Gould's and Rothe's identities using binary words, introducing a novel double-sum extension of the q-Chu-Vandermonde formula.

Contribution

It provides the first bijective proofs of Gould's and Rothe's identities and extends the q-Chu-Vandermonde formula through a new combinatorial approach.

Findings

Bijective proof of Gould's identity in binary words

Derivation of Rothe's identity via bijection from Gould's identity

Extension of the q-Chu-Vandermonde formula with a double-sum expression

Abstract

We first give a bijective proof of Gould's identity in the model of binary words. Then we deduce Rothe's identity from Gould's identity again by a bijection, which also leads to a double-sum extension of the $q$-Chu-Vandermonde formula.