probabilists' Hermite polynomials

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:orthogonal_polynomials gptkb:mathematical_concept
gptkbp:category	gptkb: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!}
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
https://www.w3.org/2000/01/rdf-schema#label	probabilists' Hermite polynomials