Sun's curious identity

In combinatorics, Sun's curious identity is the following identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002:

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

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" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 4: A05, arXiv:math.CO/0401216, Bibcode:2004math......1216C.

• Chu, W.; Claudio, L.V.D. (2003), "Jensen proof of a curious binomial identity" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 3: A20.

• Ekhad, S. B.; Mohammed, M. (2003), "A WZ proof of a 'curious' identity" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 3: A06.

• Merlini, D.; Sprugnoli, R. (2002), "A Riordan array proof of a curious identity" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2: A08.

• Panholzer, A.; Prodinger, H. (2002), "A generating functions proof of a curious identity" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2: A06.

• Sun, Zhi-Wei (2002), "A curious identity involving binomial coefficients" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2: A04.

• Sun, Zhi-Wei (2008), "On sums of binomial coefficients and their applications", Discrete Mathematics, 308 (18): 4231–4245, arXiv:math.NT/0404385, doi:10.1016/j.disc.2007.08.046, S2CID 14089498.