Hermite differential operator

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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Writing Hermite’s differential equation (see Tables 18.3.1 and 18.8.1) in the form above, the eigenfunctions are ${\mathrm{e}}^{-x^2/2}{H}_n\left(x\right)$ ($H_n$ a Hermite polynomial, $n=0,1,2,\mathrm{\dots }$), with eigenvalues ${\lambda }_n=2n+1\in {𝝈}_p$, for the differential operator … ►

1.18.43 $$f(x)=\sum _{n=0}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-(x^2+y^2)/2}}{{\pi }^{\frac{1}{2}}{2}^{n}n!}{H}_n\left(x\right)H_n\left(y\right)f(y)dy,$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral and $n$: nonnegative integer

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http://dlmf.nist.gov/1.18.E43

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pMML, png, TeX

See also:

Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

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2: Bibliography R

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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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3: 18.27 $q$-Hahn Class

… ►In the $q$-Hahn class OP’s the role of the operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the $q$-derivative ${𝒟}_q$, as defined in (17.2.41). …

4: 18.3 Definitions

§18.3 Definitions

►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►

As eigenfunctions of second order differential operators (Bochner’s theorem, Bochner (1929)). See the differential equations $A(x)p_n^{\prime \prime }(x)+B(x)p_n^{\prime }(x)+{\lambda }_n{p}_n(x)=0$, in Table 18.8.1.

… ►Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of $x-1$ for Jacobi polynomials, in powers of $x$ for the other cases). …

5: 18.38 Mathematical Applications

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Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►A further operator, the so-called Casimir operator … ►

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

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6: Bibliography K

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Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

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External Links:

FullText, MathReview Entry, ZentralBlatt

Referenced by:

§18.9(iii).

Encodings:

BibTeX

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

BibTeX

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Zeros of exceptional Hermite polynomials. J. Approx. Theory 200, pp. 28–39.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Mario Pérez Riera)

Referenced by:

§18.39(i).

Encodings:

BibTeX

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Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§18.36(i).

Encodings:

BibTeX

7: Errata

… ►This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions. … ►The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. … ►

2.3.6 $${𝒱}_{a,b}\left(f(t)\right)={\int }_a^b\left|f^{\prime }(t)\right|dt$$

The integrand has been corrected so that the absolute value does not include the differential.

Reported by Juan Luis Varona on 2021-02-08

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1.4.34 $${𝒱}_{a,b}\left(f\right)={\int }_a^b\left|f^{\prime }(x)\right|dx$$

The integrand has been corrected so that the absolute value does not include the differential.

Reported by Tran Quoc Viet on 2020-08-11

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Several changes have been made to

1. make consistent use of the Fourier transform notations $ℱ\left(f\right)$, $ℱ\left(\varphi \right)$ and $ℱ\left(u\right)$ where $f$ is a function of one real variable, $\varphi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

2. introduce the partial differential operator $𝐃$ in (1.16.30);

3. clarify the definition (1.16.32) of the partial differential operator $P(𝐃)$; and

4. clarify the use of $P(𝐃)$ and $P(𝐱)$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

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8: 18.36 Miscellaneous Polynomials

… ►They are related to Hermite–Padé approximation and can be used for proofs of irrationality or transcendence of interesting numbers. … ►

Type III $X_2$-Hermite EOP’s

►Hermite EOP’s are defined in terms of classical Hermite OP’s. The type III $X_2$-Hermite EOP’s, missing polynomial orders $1$ and $2$, are the complete set of polynomials, with real coefficients and defined explicitly as … ►Completeness and orthogonality follow from the self-adjointness of the corresponding Schrödinger operator, Gómez-Ullate and Milson (2014), Marquette and Quesne (2013).

9: 18.39 Applications in the Physical Sciences

… ►The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. … ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►c) A Rational SUSY Potential argument … ►Analogous to (18.39.7) the 3D Schrödinger operator is … ►The radial operator (18.39.28) …

10: 1.17 Integral and Series Representations of the Dirac Delta

… ►for all functions $\varphi (x)$ that are continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►More generally, assume $\varphi (x)$ is piecewise continuous (§1.4(ii)) when $x\in [-c,c]$ for any finite positive real value of $c$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►provided that $\varphi (x)$ is continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$ (as in the case of (1.17.6)). … ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. … ►

Hermite Polynomials (§18.3)

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