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:

$(x+m+1)\sum _{i=0}^m(-1{)}^i(\genfrac{}{}{0ex}{}{x+y+i}{m-i})(\genfrac{}{}{0ex}{}{y+2i}{i})-\sum _{i=0}^m(\genfrac{}{}{0ex}{}{x+i}{m-i})(-4{)}^i=(x-m)(\genfrac{}{}{0ex}{}{x}{m}).$

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

• 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 .