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gptkbp:instanceOf
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gptkb:orthogonal_polynomials
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gptkbp:category
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gptkb:software
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gptkbp:definedIn
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gptkb:Rodrigues'_formula
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gptkbp:degree
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n
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gptkbp:differential
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(1-x^2)y'' - 2xy' + n(n+1)y = 0
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gptkbp:field
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gptkb:mathematics
mathematical physics
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gptkbp:firstFew
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P_0(x) = 1
P_1(x) = x
P_2(x) = (3x^2 - 1)/2
P_3(x) = (5x^3 - 3x)/2
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gptkbp:generatingFunction
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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
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gptkb:Adrien-Marie_Legendre
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gptkbp:orthogonalityRelation
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\int_{-1}^1 P_m(x)P_n(x)dx = 2/(2n+1) \delta_{mn}
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gptkbp:orthogonalOn
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interval [-1, 1]
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gptkbp:par
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P_n(-x) = (-1)^n P_n(x)
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gptkbp:real
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yes
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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:relatedTo
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gptkb:Gegenbauer_polynomials
gptkb:Jacobi_polynomials
gptkb:Chebyshev_polynomials
associated Legendre polynomials
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gptkbp:RodriguesFormula
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P_n(x) = (1/2^n n!) d^n/dx^n (x^2-1)^n
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gptkbp:roots
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all real and in (-1,1)
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gptkbp:satisfies
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gptkb:Legendre_differential_equation
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gptkbp:symbol
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P_n(x)
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gptkbp:type
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classical polynomial
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gptkbp:usedIn
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numerical analysis
approximation theory
spherical harmonics
solution of differential equations
Gauss-Legendre quadrature
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gptkbp:bfsParent
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gptkb:Hermite_polynomial
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gptkbp:bfsLayer
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7
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https://www.w3.org/2000/01/rdf-schema#label
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Legendre polynomial
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