A partition identity on binomial coefficients and its applications (Q1119585)

The main result of the author in this paper is a useful binomial identity concerning two independent variables which is arrived at by invoking simple power series technique. Several known partition identities are listed as a consequence of the main result stated as follows: Let x and y be the complex numbers. Then \[ \left( \begin{matrix} xy\\ n\end{matrix} \right)=\sum_{\sigma (n)}\left( \begin{matrix} x\\ \bar k\end{matrix} \right)\prod \left( \begin{matrix} y\\ i\end{matrix} \right)^{k_ i\quad}, \] where \(\sigma\) (n) denotes the set of partitions of the nonnegative integer n, \(\bar k=(k_ 1,...,k_ n)\), \(\prod\) runs over i from 1 to n, and \(\left( \begin{matrix} x\\ \bar k\end{matrix} \right)\) represents the multinomial coefficient.