mathematics

Sun's curious identity

Sun's curious identity is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Sun's curious identity rather than just read about it. In short: 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}{…

Key takeaways

Sun's curious identity belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sun's curious identity to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sun's curious identity from memory before moving on to harder problems.

Reference excerpt

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.

References Callan, D. (2004), "A combinatorial proof of Sun's 'curious' identity" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 4: A05, arXiv:math.CO/0401216, Bibcode:2004math......1216C. Chu, W.; Claudio, L.V.D. (2003), "Jensen proof of a curious binomial identity" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 3: A20. Ekhad, S. B.; Mohammed, M. (2003), "A WZ proof of a 'curious' identity" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 3: A06. Merlini, D.; Sprugnoli, R. (2002), "A Riordan array proof of a curious identity" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2: A08. Panholzer, A.; Prodinger, H. (2002), "A generating functions proof of a curious identity" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2: A06. Sun, Zhi-Wei (2002), "A curious identity involving binomial coefficients" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2: A04. Sun, Zhi-Wei (2008), "On sums of binomial coefficients and their applications", Discrete Mathematics, 308 (18): 4231–4245, arXiv:math.NT/0404385, doi:10.1016/j.disc.2007.08.046, S2CID 14089498.

Worked examples

Example 1 — a first encounter with Sun's curious identity

Start with the simplest possible case. Write down what Sun's curious identity claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sun's curious identity before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sun's curious identity ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sun's curious identity

In research: Sun's curious identity appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sun's curious identity in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sun's curious identity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic identities, Factorial and binomial topics, so understanding it makes those chapters shorter.
In everyday life: Look for Sun's curious identity outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Sun's curious identity in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sun's curious identity means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Sun's curious identity out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sun's curious identity in simple terms?

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 _{…

Why does Sun's curious identity matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sun's curious identity?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sun's curious identity.