generalized binomial theorem
E27434
The generalized binomial theorem extends the classical binomial theorem by allowing real or complex exponents, expressing powers of a binomial as an infinite series using generalized binomial coefficients.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Newton binomial formula | 1 |
| Newton's generalized binomial theorem | 1 |
| generalized binomial theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T209667 — 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: generalized binomial theorem Context triple: [binomial theorem, hasGeneralization, generalized binomial theorem]
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
- 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: generalized binomial theorem Target entity description: The generalized binomial theorem extends the classical binomial theorem by allowing real or complex exponents, expressing powers of a binomial as an infinite series using generalized binomial coefficients.
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in analysis ⓘ |
| allowsExponentType |
complex exponents
ⓘ
real exponents ⓘ |
| alsoKnownAs |
binomial series
ⓘ
binomial series expansion ⓘ |
| assumes | principal branch of complex power for (1+z)^α ⓘ |
| category | series expansion theorem ⓘ |
| coefficientDefinition |
(α choose k) = Γ(α+1)/(Γ(k+1)Γ(α-k+1)) when defined
ⓘ
(α choose k) = α(α-1)…(α-k+1)/k! ⓘ |
| convergenceCondition | |z| < 1 for general complex α ⓘ |
| expresses | (1+z)^α as an infinite series ⓘ |
| extends | binomial theorem ⓘ |
| field |
algebra
ⓘ
combinatorics ⓘ complex analysis ⓘ mathematical analysis ⓘ |
| generalizes | finite binomial expansion to infinite series ⓘ |
| givesSeriesFor |
(1+z)^α
ⓘ
(1-z)^{-α} ⓘ |
| historicalAttribution | Isaac Newton ⓘ |
| historicalPeriod | 17th century ⓘ |
| implies |
analyticity of (1+z)^α on unit disk minus branch cut
ⓘ
radius of convergence 1 for binomial series in z ⓘ |
| mainFormula | (1+z)^α = Σ_{k=0}^{∞} (α choose k) z^k ⓘ |
| relatedConcept |
Gamma function
ⓘ
Pochhammer symbol ⓘ Taylor series ⓘ hypergeometric series ⓘ power series expansion ⓘ |
| requires | |arg(1+z)| < π for standard complex branch ⓘ |
| specialCase | reduces to classical binomial theorem when α is a nonnegative integer ⓘ |
| topicOf |
advanced calculus courses
ⓘ
complex analysis courses ⓘ generating function methods in combinatorics ⓘ real analysis courses ⓘ |
| usedIn |
analytic continuation of (1+z)^α
ⓘ
asymptotic expansions ⓘ probability theory ⓘ series solutions of differential equations ⓘ |
| usedToDerive |
series for (1+z)^{1/2}
ⓘ
series for (1-z)^{-1/2} ⓘ series for (1-z)^{-1} = Σ_{k=0}^{∞} z^k ⓘ |
| uses | generalized binomial coefficients ⓘ |
| validFor |
complex exponent α
ⓘ
complex variable z ⓘ |
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: generalized binomial theorem Description of subject: The generalized binomial theorem extends the classical binomial theorem by allowing real or complex exponents, expressing powers of a binomial as an infinite series using generalized binomial coefficients.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Newton's generalized binomial theorem
this entity surface form:
Newton binomial formula