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gptkbp:instanceOf
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gptkb:mathematical_concept
orthogonal polynomials
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gptkbp:application
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gptkb:Gaussian_quadrature
gptkb:physicists'_Hermite_polynomials
gptkb:probabilists'_Hermite_polynomials
solution to quantum harmonic oscillator
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gptkbp:category
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special functions
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gptkbp:degree
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n
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gptkbp:differential
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y'' - 2x y' + 2n y = 0
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gptkbp:domain
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real numbers
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gptkbp:firstFewPolynomials
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H_0(x) = 1
H_1(x) = 2x
H_2(x) = 4x^2 - 2
H_3(x) = 8x^3 - 12x
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gptkbp:generatingFunction
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exp(2xt - t^2) = sum_{n=0}^∞ H_n(x) t^n / n!
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https://www.w3.org/2000/01/rdf-schema#label
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Hermite polynomial
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gptkbp:namedAfter
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gptkb:Charles_Hermite
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gptkbp:orthogonalWithRespectTo
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weight function exp(-x^2)
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gptkbp:recurrence
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H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x)
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gptkbp:relatedTo
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gptkb:Chebyshev_polynomial
gptkb:Laguerre_polynomial
gptkb:Legendre_polynomial
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gptkbp:usedIn
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gptkb:probability_theory
numerical analysis
quantum mechanics
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gptkbp:variant
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x
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gptkbp:bfsParent
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gptkb:parabolic_cylinder_function
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gptkbp:bfsLayer
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6
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