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gptkbp:instanceOf
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orthogonal polynomials
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gptkbp:alsoKnownAs
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gptkb:ultraspherical_polynomials
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gptkbp:definedIn
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interval [-1, 1]
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gptkbp:degree
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n (non-negative integer)
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gptkbp:domainOfOrthogonality
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[-1, 1]
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gptkbp:first_terms
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C_0^{(λ)}(x) = 1
C_1^{(λ)}(x) = 2λx
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gptkbp:generatingFunction
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(1-2xt+t^2)^{-λ} = \\sum_{n=0}^\\infty C_n^{(λ)}(x)t^n
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gptkbp:hasSpecialCase
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gptkb:Jacobi_polynomials
gptkb:Legendre_polynomials
gptkb:Chebyshev_polynomials
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https://www.w3.org/2000/01/rdf-schema#label
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Gegenbauer polynomials
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gptkbp:lambdaRestriction
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λ > -1/2
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gptkbp:namedAfter
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gptkb:Leopold_Gegenbauer
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gptkbp:orthogonalityCondition
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\\int_{-1}^1 (1-x^2)^{λ-1/2} C_m^{(λ)}(x) C_n^{(λ)}(x) dx = 0, m \\neq n
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gptkbp:orthogonalWithRespectTo
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weight function (1-x^2)^{λ-1/2}
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gptkbp:parameter
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lambda (λ)
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gptkbp:publishedIn
|
gptkb:Crelle's_Journal_(1885)
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gptkbp:recurrence
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(n+1)C_{n+1}^{(λ)}(x) = 2(n+λ)xC_n^{(λ)}(x) - (n+2λ-1)C_{n-1}^{(λ)}(x)
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gptkbp:relatedTo
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Fourier series
spherical harmonics
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gptkbp:satisfies
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gptkb:Gegenbauer_differential_equation
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gptkbp:usedIn
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harmonic analysis
mathematical physics
approximation theory
solution of Laplace's equation in hyperspherical coordinates
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gptkbp:bfsParent
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gptkb:Jacobi_polynomials
gptkb:Chebyshev_polynomials
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gptkbp:bfsLayer
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7
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