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gptkbp:instanceOf
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gptkb:orthogonal_polynomials
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gptkbp:category
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special functions
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gptkbp:definedIn
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interval [-1, 1]
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gptkbp:degree
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n (where n is a non-negative integer)
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gptkbp:firstPolynomial
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P_0(x) = 1
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gptkbp:form
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gptkb:sequence_of_polynomials
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gptkbp:haveGeneratingFunction
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(1-2xt+t^2)^{-1/2} = \sum_{n=0}^\infty P_n(x)t^n
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gptkbp:namedAfter
|
gptkb:Adrien-Marie_Legendre
|
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gptkbp:orthogonal
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with respect to the L2 inner product
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gptkbp:orthogonalWithRespectTo
|
weight function 1
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gptkbp:recurrence
|
(n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x)
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gptkbp:satisfies
|
gptkb:Legendre_differential_equation
|
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gptkbp:secondPolynomial
|
P_1(x) = x
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gptkbp:symbol
|
P_n(x)
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gptkbp:thirdPolynomial
|
P_2(x) = (3x^2 - 1)/2
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gptkbp:usedIn
|
gptkb:Gauss–Legendre_quadrature
numerical analysis
physics
approximation theory
solution of Laplace's equation
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|
gptkbp:bfsParent
|
gptkb:Turán's_inequalities
gptkb:Chebyshev_polynomials
gptkb:Gauss–Legendre_quadrature
gptkb:Adrien-Marie_Legendre
gptkb:Chebyshev_Polynomials
gptkb:Hermite_polynomials
gptkb:Sturm-Liouville_theory
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gptkbp:bfsLayer
|
7
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|
https://www.w3.org/2000/01/rdf-schema#label
|
Legendre polynomials
|