Gegenbauer polynomials
E697761
Gegenbauer polynomials are a family of orthogonal polynomials on the interval [-1, 1] that generalize Legendre polynomials and play a key role in harmonic analysis and solutions of differential equations with spherical symmetry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Gegenbauer polynomials canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871819 — 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: Gegenbauer polynomials Context triple: [Jacobi polynomials, generalizes, Gegenbauer polynomials]
-
A.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
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B.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
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C.
Bernoulli polynomials
Bernoulli polynomials are a sequence of polynomials deeply connected to number theory and analysis, appearing in the study of special functions, series expansions, and the evaluation of sums of powers of integers.
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D.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
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E.
Hermite functions
Hermite functions are a family of orthogonal functions built from Hermite polynomials and a Gaussian weight, widely used in quantum mechanics, signal processing, and approximation theory.
- 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: Gegenbauer polynomials Target entity description: Gegenbauer polynomials are a family of orthogonal polynomials on the interval [-1, 1] that generalize Legendre polynomials and play a key role in harmonic analysis and solutions of differential equations with spherical symmetry.
-
A.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
-
B.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
-
C.
Bernoulli polynomials
Bernoulli polynomials are a sequence of polynomials deeply connected to number theory and analysis, appearing in the study of special functions, series expansions, and the evaluation of sums of powers of integers.
-
D.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
-
E.
Hermite functions
Hermite functions are a family of orthogonal functions built from Hermite polynomials and a Gaussian weight, widely used in quantum mechanics, signal processing, and approximation theory.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
family of orthogonal polynomials
ⓘ
special functions ⓘ |
| alsoKnownAs | ultraspherical polynomials ⓘ |
| associatedWith | dimension parameter d via \lambda = (d-2)/2 ⓘ |
| belongTo | Askey scheme of hypergeometric orthogonal polynomials NERFINISHED ⓘ |
| definedOn | interval [-1,1] ⓘ |
| degree | n in variable x ⓘ |
| denotedBy | C_n^{(\lambda)}(x) NERFINISHED ⓘ |
| dependOn | degree n ⓘ |
| expressibleAs | hypergeometric function {}_2F_1 ⓘ |
| field | mathematics ⓘ |
| generalize |
Chebyshev polynomials
NERFINISHED
ⓘ
Legendre polynomials NERFINISHED ⓘ |
| haveGeneratingFunction | (1-2xt+t^2)^{-\lambda} = \sum_{n=0}^{\infty} C_n^{(\lambda)}(x) t^n ⓘ |
| leadingCoefficient | 2^n \frac{\Gamma(n+\lambda)}{n!\,\Gamma(\lambda)} ⓘ |
| namedAfter | Leopold Gegenbauer NERFINISHED ⓘ |
| orthogonalityCondition | \lambda > -1/2 ⓘ |
| orthogonalityIntegral | \int_{-1}^1 (1-x^2)^{\lambda-1/2} C_m^{(\lambda)}(x) C_n^{(\lambda)}(x) dx = 0 for m \neq n ⓘ |
| orthogonalOn | [-1,1] ⓘ |
| orthogonalWithRespectTo | weight function (1-x^2)^{\lambda-1/2} ⓘ |
| parameter | \lambda ⓘ |
| recurrenceRelation | (n+1)C_{n+1}^{(\lambda)}(x) = 2(n+\lambda)x C_n^{(\lambda)}(x) - (n+2\lambda-1)C_{n-1}^{(\lambda)}(x) ⓘ |
| relatedTo | spherical harmonics on S^{d-1} ⓘ |
| satisfy |
second-order linear differential equation
ⓘ
three-term recurrence relation ⓘ |
| satisfyDifferentialEquation | (1-x^2)y'' - (2\lambda+1)xy' + n(n+2\lambda)y = 0 ⓘ |
| specialCase |
Chebyshev polynomials of the first kind for \lambda = 0 (limit case)
NERFINISHED
ⓘ
Chebyshev polynomials of the second kind for \lambda = 1 NERFINISHED ⓘ Legendre polynomials for \lambda = 1/2 ⓘ |
| subfield |
harmonic analysis
ⓘ
mathematical physics ⓘ orthogonal polynomials ⓘ special functions ⓘ |
| symmetricProperty | C_n^{(\lambda)}(-x) = (-1)^n C_n^{(\lambda)}(x) ⓘ |
| usedFor |
addition theorems for spherical harmonics
ⓘ
expansion of powers of distance in higher dimensions ⓘ |
| usedIn |
approximation theory
ⓘ
expansion of functions on spheres ⓘ harmonic analysis ⓘ potential theory ⓘ quantum mechanics with central potentials ⓘ solutions of differential equations with spherical symmetry ⓘ spectral methods for partial differential equations ⓘ |
| valueAt |
C_n^{(\lambda)}(0) = 0 for odd n
ⓘ
C_n^{(\lambda)}(1) = \binom{n+2\lambda-1}{n} ⓘ C_{2k}^{(\lambda)}(0) = (-1)^k \frac{\Gamma(k+\lambda)}{k!\,\Gamma(\lambda)} ⓘ |
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Instruction
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Input
Subject: Gegenbauer polynomials Description of subject: Gegenbauer polynomials are a family of orthogonal polynomials on the interval [-1, 1] that generalize Legendre polynomials and play a key role in harmonic analysis and solutions of differential equations with spherical symmetry.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.