physicists' Hermite polynomials

URI: https://gptkb.org/entity/physicists'_Hermite_polynomials
GPTKB entity
Statements (23)
Predicate Object
gptkbp:instanceOf gptkb:orthogonal_polynomials
gptkbp:alsoKnownAs Hermite polynomials of the first kind
gptkbp:application gptkb:Gaussian_quadrature
solution to quantum harmonic oscillator
expansion of functions in terms of orthogonal polynomials
gptkbp:differential y'' - 2x y' + 2n y = 0
gptkbp:distinctFrom probabilists' Hermite polynomials differ by scaling factor
gptkbp:domain real numbers
gptkbp:firstFewPolynomials H_0(x) = 1
H_1(x) = 2x
H_2(x) = 4x^2 - 2
H_3(x) = 8x^3 - 12x
gptkbp:generatingFunction exp(2xt - t^2) = sum_{n=0}^∞ H_n(x) t^n / n!
gptkbp:namedAfter gptkb:Charles_Hermite
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:probabilists'_Hermite_polynomials
gptkbp:usedIn gptkb:probability_theory
quantum mechanics
gptkbp:bfsParent gptkb:Hermite_polynomial
gptkbp:bfsLayer 7
https://www.w3.org/2000/01/rdf-schema#label physicists' Hermite polynomials