Sun's curious identity explained

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 ^{m( -1 )} _{i =0} ^{m} ^{i =0} \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

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