Sun's curious identity

Template:Short description 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

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