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

10.37236/462 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Victor J. W. Guo ◽  
Jiang Zeng
Keyword(s):  
Binomial Coefficient ◽  
Combinatorial Proof ◽  
Binomial Coefficient Identity

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].

scholarly journals A $q$-Analogue of some Binomial Coefficient Identities of Y. Sun

10.37236/565 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Victor J. W. Guo ◽  
Dan-Mei Yang
Keyword(s):  
Binomial Coefficient

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.

Integral Functions Associated with Certain Binomial Coefficient Sums

1936 ◽  
Vol 43 (1) ◽  
pp. 27-32
Author(s):  
George Rutledge ◽  
R. D. Douglass
Keyword(s):  
Binomial Coefficient

scholarly journals Analogues of the binomial coefficient theorems of Gauss and Jacobi

2016 ◽  
Vol 12 (08) ◽  
pp. 2125-2145
Author(s):  
Abdullah Al-Shaghay ◽  
Karl Dilcher
Keyword(s):  
Gamma Function ◽  
Simple Form ◽  
Binomial Coefficient ◽  
Catalan Numbers ◽  
Binomial Coefficients

The theorems of Gauss and Jacobi that give modulo [Formula: see text] evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo [Formula: see text]. In this paper, we further extend these results to congruences modulo [Formula: see text]. In the process, we prove congruences to arbitrarily high powers of [Formula: see text] for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's [Formula: see text]-adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients.

A Binomial Coefficient Summation (Gengzhe Chang)

SIAM Review ◽  
1983 ◽  
Vol 25 (4) ◽  
pp. 575-576
Author(s):  
J. Roppert
Keyword(s):  
Binomial Coefficient
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