Sun's curious identity
Encyclopedia
In combinatorics
, Sun's curious identity is the following identity
involving binomial coefficient
s, first established by Zhi-Wei Sun
in 2002:
After Sun's publication of this identity, five other proofs were obtained by various mathematicians: they are 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, and Callan's combinatorial proof
involving dominos and colorings.
An application of the identity was given by Sun in a recent paper.
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, Sun's curious identity is the following identity
Identity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
involving binomial coefficient
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
s, first established by Zhi-Wei Sun
Zhi-Wei Sun
Zhi-Wei Sun is a Chinese mathematician, working primarily on number theory, combinatorics, and group theory. Currently he works as a professor in Nanjing University....
in 2002:
After Sun's publication of this identity, five other proofs were obtained by various mathematicians: they are Panholzer and Prodinger's proof via generating functions, Merlini and Sprugnoli's proof using Riordan
John Riordan
John Riordan was an American mathematician and the author of major early works in combinatorics, particularly Introduction to Combinatorial Analysis and Combinatorial Identities.- Life :...
arrays, Ekhad and Mohammed's proof by the WZ method, Chu and Claudio's proof with the help of Jensen's formula, and Callan's combinatorial proof
Combinatorial proof
In mathematics, the term combinatorial proof is often used to mean either of two types of proof of an identity in enumerative combinatorics that either states that two sets of combinatorial configurations, depending on one or more parameters, have the same number of elements , or gives a formula...
involving dominos and colorings.
An application of the identity was given by Sun in a recent paper.