Sun's curious identity
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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:
- [math]\displaystyle{ (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}. }[/math]
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
- Callan, D. (2004), "A combinatorial proof of Sun's 'curious' identity", INTEGERS: The Electronic Journal of Combinatorial Number Theory 4: A05, Bibcode: 2004math......1216C, http://www.emis.de/journals/INTEGERS/papers/e5/e5.pdf.
- Chu, W.; Claudio, L.V.D. (2003), "Jensen proof of a curious binomial identity", INTEGERS: The Electronic Journal of Combinatorial Number Theory 3: A20, http://www.emis.de/journals/INTEGERS/papers/d20/d20.pdf.
- "A WZ proof of a 'curious' identity", INTEGERS: The Electronic Journal of Combinatorial Number Theory 3: A06, 2003, http://www.emis.de/journals/INTEGERS/papers/d6/d6.pdf.
- Merlini, D.; Sprugnoli, R. (2002), "A Riordan array proof of a curious identity", INTEGERS: The Electronic Journal of Combinatorial Number Theory 2: A08, http://www.emis.de/journals/INTEGERS/papers/c8/c8.pdf.
- Panholzer, A.; Prodinger, H. (2002), "A generating functions proof of a curious identity", INTEGERS: The Electronic Journal of Combinatorial Number Theory 2: A06, http://www.emis.de/journals/INTEGERS/papers/c6/c6.pdf.
- Sun, Zhi-Wei (2002), "A curious identity involving binomial coefficients", INTEGERS: The Electronic Journal of Combinatorial Number Theory 2: A04, http://www.emis.de/journals/INTEGERS/papers/c4/c4.pdf.
- Sun, Zhi-Wei (2008), "On sums of binomial coefficients and their applications", Discrete Mathematics 308 (18): 4231–4245, doi:10.1016/j.disc.2007.08.046.
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