A \(q\)-analogue of some binomial coefficient identities of Y. Sun
The electronic journal of combinatorics, Tome 18 (2011) no. 1
We give a $q$-analogue of some binomial coefficient identities of Y. Sun [Electron. J. Combin. 17 (2010), #N20] as follows: \begin{align*} \sum_{k=0}^{\lfloor n/2\rfloor}{m+k\brack k}_{q^2}{m+1\brack n-2k}_{q} q^{n-2k\choose 2} &={m+n\brack n}_{q}, \\ \sum_{k=0}^{\lfloor n/4\rfloor}{m+k\brack k}_{q^4}{m+1\brack n-4k}_{q} q^{n-4k\choose 2} &=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k{m+k\brack k}_{q^2}{m+n-2k\brack n-2k}_{q}, \end{align*} where ${n\brack k}_q$ stands for the $q$-binomial coefficient. We provide two proofs, one of which is combinatorial via partitions.
@article{10_37236_565,
author = {Victor J. W. Guo and Dan-Mei Yang},
title = {A \(q\)-analogue of some binomial coefficient identities of {Y.} {Sun}},
journal = {The electronic journal of combinatorics},
year = {2011},
volume = {18},
number = {1},
doi = {10.37236/565},
zbl = {1230.05012},
url = {http://geodesic.mathdoc.fr/articles/10.37236/565/}
}
Victor J. W. Guo; Dan-Mei Yang. A \(q\)-analogue of some binomial coefficient identities of Y. Sun. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/565
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