An involution for the Gauss identity (Q1395818)

Several proofs of the Gauss identity \[ \sum^m_{r=0} (-1)^r {m\brack r}_q= \begin{cases} 0\quad & (m\text{ odd})\\ (1- q)(1- q^3)\cdots(1- q^{m- 1})\quad &(m\text{ even})\end{cases} \] have been given by various authors. Here a combinatorial proof is given, based on Ferrers diagrams. Further, a refinement of the approach leads to a generalisation of the Gauss identity, namely \[ \sum^m_{r=0} (-1)^r {n+r\brack n}_q {n+m-r\brack n}_q= \begin{cases} 0\quad &(m\text{ odd})\\ {n+m/2\brack n}_{q^2}\quad &(m\text{ even}).\end{cases} \] Gauss's identity is obtained from it by letting \(n\to\infty\). Finally, Gauss's identity is extended to evaluate \[ \sum \zeta^{r_1+ 2r_2+\cdots+ pr_p} {n+r_1\brack n}_q\cdots {n+r_p\brack n}_q \] where the sum is over all \((r_1,\dots, r_p)\) such that \(r_1+\cdots+ r_p= m\), and where \(\zeta= e^{2\pi i/p}\) is the \(p\)th root of unity.