Sun's curious identity
{{short description|Identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002}}
In combinatorics, Sun's curious identity is the following identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002:
:
(x+m+1)\sum_{i=0}^m(-1)^i\dbinom{x+y+i}{m-i}\dbinom{y+2i}{i}
-\sum_{i=0}^{m}\dbinom{x+i}{m-i}(-4)^i=(x-m)\dbinom{x}{m}.
Proofs
After Sun's publication of this identity in 2002, five other proofs were obtained by various mathematicians:
- Panholzer and Prodinger's proof via generating functions;
- Merlini and Sprugnoli's proof using Riordan arrays;
- Ekhad and Mohammed's proof by the WZ method;
- Chu and Claudio's proof with the help of Jensen's formula;
- Callan's combinatorial proof involving dominos and colorings.
References
- {{citation
| last = Callan | first = D.
| arxiv = math.CO/0401216
| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory
| page = A05
| title = A combinatorial proof of Sun's 'curious' identity
| url = http://www.emis.de/journals/INTEGERS/papers/e5/e5.pdf
| volume = 4
| year = 2004| bibcode = 2004math......1216C
}}.
- {{citation
| last1 = Chu | first1 = W.
| last2 = Claudio | first2 = L.V.D.
| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory
| page = A20
| title = Jensen proof of a curious binomial identity
| url = http://www.emis.de/journals/INTEGERS/papers/d20/d20.pdf
| volume = 3
| year = 2003}}.
- {{citation
| last1 = Ekhad | first1 = S. B. | author1-link = Doron Zeilberger
| last2 = Mohammed | first2 = M.
| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory
| page = A06
| title = A WZ proof of a 'curious' identity
| url = http://www.emis.de/journals/INTEGERS/papers/d6/d6.pdf
| volume = 3
| year = 2003}}.
- {{citation
| last1 = Merlini | first1 = D.
| last2 = Sprugnoli | first2 = R.
| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory
| page = A08
| title = A Riordan array proof of a curious identity
| url = http://www.emis.de/journals/INTEGERS/papers/c8/c8.pdf
| volume = 2
| year = 2002}}.
- {{citation
| last1 = Panholzer | first1 = A.
| last2 = Prodinger | first2 = H.
| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory
| page = A06
| title = A generating functions proof of a curious identity
| url = http://www.emis.de/journals/INTEGERS/papers/c6/c6.pdf
| volume = 2
| year = 2002}}.
- {{citation
| last = Sun | first = Zhi-Wei
| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory
| page = A04
| title = A curious identity involving binomial coefficients
| url = http://www.emis.de/journals/INTEGERS/papers/c4/c4.pdf
| volume = 2
| year = 2002}}.
- {{citation
| last = Sun | first = Zhi-Wei
| doi = 10.1016/j.disc.2007.08.046
| arxiv = math.NT/0404385
| issue = 18
| journal = Discrete Mathematics
| pages = 4231–4245
| title = On sums of binomial coefficients and their applications
| volume = 308
| year = 2008| s2cid = 14089498
}}.