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gptkbp:instanceOf
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gptkb:mathematical_concept
orthogonal polynomials
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gptkbp:category
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special functions
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gptkbp:definedIn
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real line
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gptkbp:differential
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y'' - 2x y' + 2n y = 0
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gptkbp:field
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gptkb:mathematics
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gptkbp:firstPolynomial
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H_0(x) = 1
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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 polynomials
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gptkbp:namedAfter
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gptkb:Charles_Hermite
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gptkbp:orthogonalityRelation
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∫_{-∞}^{∞} H_m(x) H_n(x) e^{-x^2} dx = 0 for m ≠ n
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gptkbp:orthogonalOn
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(-∞, ∞)
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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:Laguerre_polynomials
gptkb:Legendre_polynomials
gptkb:Chebyshev_polynomials
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gptkbp:secondPolynomial
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H_1(x) = 2x
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gptkbp:symbol
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H_n(x)
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gptkbp:thirdPolynomial
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H_2(x) = 4x^2 - 2
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gptkbp:usedFor
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gptkb:Gaussian_quadrature
solving the quantum harmonic oscillator
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gptkbp:usedIn
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gptkb:probability_theory
numerical analysis
physics
quantum mechanics
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gptkbp:bfsParent
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gptkb:quantum_harmonic_oscillator
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gptkbp:bfsLayer
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5
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