Combinatorial proof of a curious \(q\)-binomial coefficient identity (Q2380424)

Summary: Using the algorithm Z developed by Zeilberger, we give a combinatorial proof of the following q-binomial coefficient identity \[ \sum_{k=0}^m (-1)^{m-k} \left[{m\atop k}\right] \left[{m+k\atop a}\right] (-xq^a; q)_{n+k-a} q^{\binom{k+1}{2}-mk+\binom{a}{2}} = \sum_{k=0}^n \left[{n\atop k}\right] \left[{n+k\atop a}\right] x^{m+k-a} q^{mn+\binom{k}{2}}, \] which was obtained by \textit{S.J.X. Hou} and \textit{J. Zeng} [``A \(q\)-analog of dual sequences with applications,'' Eur. J. Comb. 28, No.\,1, 214--227 (2007; Zbl 1107.05011)].