Geodesic
Combinatorial proof of a curious \(q\)-binomial coefficient identity
The electronic journal of combinatorics, Tome 17 (2010)
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Using the Algorithm Z developed by Zeilberger, we give a combinatorial proof of the following $q$-binomial coefficient identity $$ \sum_{k=0}^m(-1)^{m-k}{m\brack k}{n+k\brack a}(-xq^a;q)_{n+k-a}q^{{k+1\choose 2}-mk+{a\choose 2}} $$ $$=\sum_{k=0}^n{n\brack k}{m+k\brack a}x^{m+k-a}q^{mn+{k\choose 2}}, $$ which was obtained by Hou and Zeng [European J. Combin. 28 (2007), 214–227].
DOI : 10.37236/462
Classification : 05A17, 05A30, 33D99 
@article{10_37236_462,
     author = {Victor J. W. Guo and Jiang Zeng},
     title = {Combinatorial proof of a curious \(q\)-binomial coefficient identity},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/462},
     zbl = {1205.05014},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/462/}
}
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Victor J. W. Guo; Jiang Zeng. Combinatorial proof of a curious \(q\)-binomial coefficient identity. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/462

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