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gptkbp:instanceOf
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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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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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|
https://www.w3.org/2000/01/rdf-schema#label
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Legendre polynomials
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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
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(n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x)
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gptkbp:satisfies
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gptkb:Legendre_differential_equation
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gptkbp:secondPolynomial
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P_1(x) = x
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gptkbp:symbol
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P_n(x)
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gptkbp:thirdPolynomial
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P_2(x) = (3x^2 - 1)/2
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gptkbp:usedIn
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gptkb:Gauss–Legendre_quadrature
numerical analysis
physics
approximation theory
solution of Laplace's equation
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gptkbp:bfsParent
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gptkb:Adrien-Marie_Legendre
gptkb:Hermite_polynomials
gptkb:Generalized_Polynomial_Chaos
gptkb:Special_functions_theory
gptkb:Sturm–Liouville_problem
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gptkbp:bfsLayer
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6
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