combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

, Sun's curious identity is the following

identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...

involving

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

s, first established by

Zhi-Wei Sun Sun Zhiwei (, born October 16, 1965) is a Chinese mathematician, working primarily in number theory, combinatorics, and group theory. He is a professor at Nanjing University. Biography Sun Zhiwei was born in Huai'an, Jiangsu. Sun and his twi ...

in 2002: :

(x+m+1)\sum_^m(-1)^i\dbinom\dbinom
-\sum_^\dbinom(-4)^i=(x-m)\dbinom.

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 In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series ...

; * 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 In mathematics, the term ''combinatorial proof'' is often used to mean either of two types of mathematical proof: * A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in t ...

involving

dominos Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also ca ...

and colorings.

References

*. *. *. *. *. *. *{{citation , last = Sun , first = Zhi-Wei , doi = 10.1016/j.disc.2007.08.046 , arxiv = math.NT/0404385 , issue = 18 , journal =

Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

, pages = 4231–4245 , title = On sums of binomial coefficients and their applications , volume = 308 , year = 2008. Factorial and binomial topics Mathematical identities