physicists' Hermite polynomials
GPTKB entity
Statements (23)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
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!
|
| https://www.w3.org/2000/01/rdf-schema#label |
physicists' Hermite polynomials
|
| 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
|