probabilists' Hermite polynomials
GPTKB entity
Statements (25)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
orthogonal polynomials |
| gptkbp:category |
orthogonal polynomials
special functions |
| gptkbp:differenceFromPhysicistsHermite |
H_n^{(prob)}(x) = 2^{-n/2} H_n^{(phys)}(x/\\sqrt{2})
|
| gptkbp:differential |
H_n'(x) = n H_{n-1}(x)
|
| gptkbp:domain |
real numbers
|
| gptkbp:explicitFormula |
H_n(x) = (-1)^n e^{x^2/2} \\frac{d^n}{dx^n} e^{-x^2/2}
|
| gptkbp:firstFewPolynomials |
H_0(x) = 1
H_1(x) = x H_2(x) = x^2 - 1 H_3(x) = x^3 - 3x |
| gptkbp:generatingFunction |
e^{xt - t^2/2} = \\sum_{n=0}^\\infty H_n(x) \\frac{t^n}{n!}
|
| https://www.w3.org/2000/01/rdf-schema#label |
probabilists' Hermite polynomials
|
| gptkbp:namedAfter |
gptkb:Charles_Hermite
|
| gptkbp:orthogonalityRelation |
\\int_{-\\infty}^{\\infty} H_m(x) H_n(x) \\frac{e^{-x^2/2}}{\\sqrt{2\\pi}} dx = n! \\delta_{mn}
|
| gptkbp:orthogonalWithRespectTo |
gptkb:standard_normal_distribution
|
| gptkbp:recurrence |
H_{n+1}(x) = x H_n(x) - n H_{n-1}(x)
|
| gptkbp:relatedTo |
gptkb:physicists'_Hermite_polynomials
|
| gptkbp:usedIn |
gptkb:probability_theory
quantum mechanics stochastic processes |
| gptkbp:variant |
x
|
| gptkbp:bfsParent |
gptkb:Hermite_polynomial
|
| gptkbp:bfsLayer |
7
|