generalized binomial theorem

E27434

mathematical theorem result in analysis

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.

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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

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

binomial theorem → hasGeneralization → generalized binomial theorem ⓘ

binomial theorem → hasGeneralization → Newton's generalized binomial theorem ⓘ

linked to: generalized binomial theorem

multinomial theorem → generalizationOf → Newton binomial formula ⓘ