|
gptkbp:instanceOf
|
gptkb:orthogonal_polynomials
gptkb:mathematical_concept
|
|
gptkbp:category
|
special functions
|
|
gptkbp:definedIn
|
real line
|
|
gptkbp:differential
|
y'' - 2x y' + 2n y = 0
|
|
gptkbp:field
|
gptkb:mathematics
|
|
gptkbp:firstPolynomial
|
H_0(x) = 1
|
|
gptkbp:generatingFunction
|
exp(2xt-t^2) = sum_{n=0}^∞ H_n(x) t^n / n!
|
|
gptkbp:namedAfter
|
gptkb:Charles_Hermite
|
|
gptkbp:orthogonalityRelation
|
∫_{-∞}^{∞} H_m(x) H_n(x) e^{-x^2} dx = 0 for m ≠ n
|
|
gptkbp:orthogonalOn
|
(-∞, ∞)
|
|
gptkbp:orthogonalWithRespectTo
|
weight function exp(-x^2)
|
|
gptkbp:recurrence
|
H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x)
|
|
gptkbp:relatedTo
|
gptkb:Laguerre_polynomials
gptkb:Legendre_polynomials
gptkb:Chebyshev_polynomials
|
|
gptkbp:secondPolynomial
|
H_1(x) = 2x
|
|
gptkbp:symbol
|
H_n(x)
|
|
gptkbp:thirdPolynomial
|
H_2(x) = 4x^2 - 2
|
|
gptkbp:usedFor
|
gptkb:Gaussian_quadrature
solving the quantum harmonic oscillator
|
|
gptkbp:usedIn
|
gptkb:probability_theory
numerical analysis
physics
quantum mechanics
|
|
gptkbp:bfsParent
|
gptkb:Charles_Hermite
gptkb:quantum_harmonic_oscillator
|
|
gptkbp:bfsLayer
|
6
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
Hermite polynomials
|