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: ( x + m + 1 ) ∑ i = 0 m ( − 1 ) i ( x + y + i m − i ) ( y + 2 i i ) − ∑ i = 0 m ( x + i m − i ) ( − 4 ) i = ( x − m ) ( x m ) . {\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.