Vandermonde's identity
E167770
Vandermonde's identity is a fundamental combinatorial formula that expresses a binomial coefficient with a sum index as a sum of products of binomial coefficients, often visualized via counting arguments or generating functions.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Chu–Vandermonde identity | 2 |
| Vandermonde convolution identity | 1 |
| Vandermonde's convolution | 1 |
| Vandermonde's identity canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1470735 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Vandermonde's identity Context triple: [Pascal's identity, relatedTo, Vandermonde's identity]
-
A.
Pascal's identity
Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
-
B.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
C.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
D.
multinomial theorem
The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
-
E.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Vandermonde's identity Target entity description: Vandermonde's identity is a fundamental combinatorial formula that expresses a binomial coefficient with a sum index as a sum of products of binomial coefficients, often visualized via counting arguments or generating functions.
-
A.
Pascal's identity
Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
-
B.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
C.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
D.
multinomial theorem
The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
-
E.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
binomial identity
ⓘ
combinatorial identity ⓘ |
| appearsIn |
courses on discrete mathematics
ⓘ
courses on generating functions ⓘ elementary combinatorics textbooks ⓘ |
| canBeProvedBy |
algebraic manipulation of binomial theorem
ⓘ
combinatorial counting argument ⓘ generating functions ⓘ induction on r ⓘ |
| field |
algebra
ⓘ
combinatorics ⓘ discrete mathematics ⓘ |
| generalizesTo |
Vandermonde's identity
self-linksurface differs
ⓘ
surface form:
Chu–Vandermonde identity
|
| hasAlternativeName |
Vandermonde's identity
ⓘ
surface form:
Vandermonde convolution identity
Vandermonde's identity ⓘ
surface form:
Vandermonde's convolution
|
| hasFormula |
For integers m,n,r: sum_{k} C(m,k) C(n,r-k) = C(m+n,r)
ⓘ
\(\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}\) ⓘ \(\sum_{k} \binom{m}{k}\binom{n}{r-k} = 0\) if r<0 or r>m+n ⓘ |
| hasHistoricalPeriod | 18th century ⓘ |
| hasInterpretation |
coefficient extraction in product of binomial series
ⓘ
counts ways to choose r objects from a union of two disjoint sets of sizes m and n ⓘ expresses binomial coefficients as a discrete convolution ⓘ |
| hasSymmetryProperty | symmetric in m and n ⓘ |
| involvesConcept |
binomial coefficient
ⓘ
binomial theorem ⓘ combinatorial proof ⓘ convolution of sequences ⓘ counting argument ⓘ generating function ⓘ multiset ⓘ |
| namedAfter | Alexandre-Théophile Vandermonde ⓘ |
| namedEntityType | mathematical identity ⓘ |
| relatedTo |
Vandermonde's identity
self-linksurface differs
ⓘ
surface form:
Chu–Vandermonde identity
Pascal's identity ⓘ
surface form:
Pascal's rule
binomial theorem ⓘ convolution of binomial coefficients ⓘ hockey-stick identity ⓘ |
| usedIn |
analysis of algorithms
ⓘ
combinatorial enumeration ⓘ number theory ⓘ probability calculations involving hypergeometric distributions ⓘ simplification of binomial sums ⓘ |
| validFor |
all integers r
ⓘ
integers m,n,r with m,n \ge 0 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Vandermonde's identity Description of subject: Vandermonde's identity is a fundamental combinatorial formula that expresses a binomial coefficient with a sum index as a sum of products of binomial coefficients, often visualized via counting arguments or generating functions.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Chu–Vandermonde identity
this entity surface form:
Chu–Vandermonde identity
this entity surface form:
Vandermonde's convolution
this entity surface form:
Vandermonde convolution identity