Combinatorial Proof of a Curious $q$-Binomial Coefficient Identity

摘要

Using the Algorithm Z developed by Zeilberger, we give a combinatorial proof of the following $q$-binomial coefficient identity $$ sum_{k=0}^m(-1)^{m-k}{mrack k}{n+krack a}(-xq^a;q)_{n+k-a}q^{{k+1choose 2}-mk+{achoose 2}} $$ $$=sum_{k=0}^n{nrack k}{m+krack a}x^{m+k-a}q^{mn+{kchoose 2}}, $$ which was obtained by Hou and Zeng [European J. Combin. 28 (2007), 214–227].