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.
Aliases (2)
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 → |
Referenced by (3)
| Subject (surface form when different) | Predicate |
|---|---|
|
binomial theorem
→
binomial theorem ("Newton's generalized binomial theorem") → |
hasGeneralization |
|
multinomial theorem
("Newton binomial formula")
→
|
generalizationOf |