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

\[\sum\limits_{k=0}^{n} \left(
\begin{array}{c}
x-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y+kz \\
n-k \\
\end{array}
\right)
= \sum\limits_{k=0}^{n}
\left(
\begin{array}{c}
x+\epsilon-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y-\epsilon+kz \\
n-k \\
\end{array}
\right)
\]

and Rothe’s identity
\[\sum\limits_{k=0}^{n}\frac{x}{x-kz}
\left(
\begin{array}{c}
x-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y+kz \\
n-k \\
\end{array}
\right)
=
\left(
\begin{array}{c}
x+y \\
n \\
\end{array}
\right)\]