A Modular Binomial Coefficient Identity: 11118

2006 ◽  
Vol 113 (7) ◽  
pp. 657
Author(s):  
Vesselin Dimitrov ◽  
Robin Chapman
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficient Identity

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 Binomial Coefficient Identity Associated to a Conjecture of Beukers

10.37236/1348 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Scott Ahlgren ◽  
Shalosh B. Ekhad ◽  
Ken Ono ◽  
Doron Zeilberger
Keyword(s):  
Binomial Coefficient ◽  
Wz Method ◽  
Binomial Coefficient Identity

Using the WZ method, a binomial coefficient identity is proved. This identity is noteworthy since its truth is known to imply a conjecture of Beukers.

scholarly journals A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apéry numbers

10.37236/1856 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Wenchang Chu
Keyword(s):  
Rational Function ◽  
Binomial Coefficient ◽  
Harmonic Number ◽  
Partial Fraction ◽  
Algebraic Identity ◽  
Apéry Numbers ◽  
Partial Fraction Decomposition ◽  
Limiting Case ◽  
Binomial Coefficient Identity

By means of partial fraction decomposition, an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity, which in turn has been shown to imply Beukers' conjecture on the congruence of Apéry numbers.

scholarly journals Counting $k$-Marked Durfee Symbols

10.37236/528 ◽  
2010 ◽  
Vol 18 (1) ◽  
Author(s):  
Kağan Kurşungöz
Keyword(s):  
Generating Functions ◽  
Binomial Coefficient ◽  
Equivalence Classes ◽  
Alternative Characterization ◽  
Binomial Coefficient Identity

An alternative characterization of $k$-marked Durfee symbols defined by Andrews is given. Some identities involving generating functions of $k$-marked Durfee symbols are proven combinatorially by considering the symbols not individually, but in equivalence classes. Also, a related binomial coefficient identity is obtained in the course.

Another Binomial Coefficient Identity: 10878

2003 ◽  
Vol 110 (4) ◽  
pp. 342
Author(s):  
Pal Peter Dalyay ◽  
Said Amghibech ◽  
David Callan
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficient Identity

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.

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