Gauss–Hermite formula

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2: 18.5 Explicit Representations

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§18.5(ii) Rodrigues Formulas

… ►Related formula: … ►

Hermite

… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite

\mathit{He}_{n}

polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►

Hermite

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3: Bibliography G

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C. F. Gauss (1863) Werke. Band II. pp. 436–447 (German).

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Notes:: Reprinted by Georg Olms Verlag, Hildesheim, 1973.
External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §27.2(i).
Encodings:: BibTeX

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W. Gautschi (1968) Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:: ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)
Referenced by:: §18.39(iii).
Encodings:: BibTeX

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W. Gautschi (2016) Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.

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Notes:: Includes two MATLAB programs claiming 12D and 30D accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 4th item, §7.25(ii).
Encodings:: BibTeX

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D. Gómez-Ullate and R. Milson (2014) Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47 (1), pp. 015203, 26 pp..

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External Links:: ISSN 1751-8113, Document, Link, MathReview (Brian Simanek)
Referenced by:: §18.36(vi), §18.36(vi).
Encodings:: BibTeX

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H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:: FullText, MathReview (B. M. Stewart), ZentralBlatt
Referenced by:: §24.15(iii), §24.6.
Encodings:: BibTeX

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4: 18.38 Mathematical Applications

… ►with

H_{n}\left(x\right)

as in §18.3, satisfies the Toda equation … ►

Random Matrix Theory

►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. … ►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas. … ►Hermite EOP’s appear in solutions of a rationally modified Schrödinger equation in §18.39. …

5: 18.3 Definitions

§18.3 Definitions

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

As given by a Rodrigues formula (18.5.5).

… ►However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). … ►Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). …

6: 18.30 Associated OP’s

… ►where the generalized hypergeometric function

{{}_{4}F_{3}}

is defined by (16.2.1). … ►

§18.30(iv) Associated Hermite Polynomials

►The recursion relation for the associated Hermite polynomials, see (18.30.2), and (18.30.3), is ►

H_{-1}\left(x;c\right)=0,

… ►For Gauss’ hypergeometric function

F

see (15.2.1). …

7: Bibliography R

… ►

I. S. Reed, D. W. Tufts, X. Yu, T. K. Truong, M. T. Shih, and X. Yin (1990) Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.

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External Links:: ZentralBlatt
Referenced by:: §27.17.
Encodings:: BibTeX

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H. Rosengren (1999) Another proof of the triple sum formula for Wigner

9j

-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

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External Links:: ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt
Referenced by:: §34.6.
Encodings:: BibTeX

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R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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External Links:: ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt
Referenced by:: Profile Ranjan Roy.
Encodings:: BibTeX

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J. Rushchitsky and S. Rushchitska (2000) 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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8: Errata

… ►We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. … ►

Chapter 18 Additions

The following additions were made in Chapter 18:

Section 18.2

In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)–18.2(xii), with Equations (18.2.21)–(18.2.46),
Section 18.3

A new introduction, minor changes in the presentation, and three new paragraphs.
Section 18.5

Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).
Section 18.8

Line numbers and two extra rows were added to Table 18.8.1.
Section 18.9

Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.
Section 18.12

Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).
Section 18.14

Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).
Section 18.15

Equation (18.15.4_5).
Section 18.16

The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).
Section 18.17

Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).
Section 18.18

Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.
Section 18.19

A new introduction.
Section 18.21

Equation (18.21.13).
Section 18.25

Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.
Section 18.26

In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).
Section 18.27

Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).
Section 18.28

A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)–18.28(xi), with Equations (18.28.23)–(18.28.34).
Section 18.30

Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)–18.30(viii), with Equations (18.30.8)–(18.30.31).
Section 18.32

This short section has been expanded, with Equation (18.32.2).
Section 18.33

Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).
Section 18.34

This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).
Section 18.35

This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).
Section 18.36

This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.
Section 18.38

The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).
Section 18.39

This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

The original text of 18.39(i) Quantum Mechanics was:

“Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass $m$ with potential energy $V(x)$ :

errata.1 $\left(\frac{-\hbar^{2}}{2m}\frac{{\partial}^{2}}{{\partial x}^{2}}+V(x)\right)% \psi(x,t)=i\hbar\frac{\partial}{\partial t}\psi(x,t),$

where $\hbar$ is the reduced Planck’s constant. On substituting $\psi(x,t)=\eta(x)\zeta(t)$ , we obtain two ordinary differential equations, each of which involve the same constant $E$ . The equation for $\eta(x)$ is

errata.2 $\frac{{\mathrm{d}}^{2}\eta}{{\mathrm{d}x}^{2}}+\frac{2m}{\hbar^{2}}\left(E-V(x% )\right)\eta=0.$

For a harmonic oscillator, the potential energy is given by

errata.3 $V(x)=\tfrac{1}{2}m\omega^{2}x^{2},$

where $\omega$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval $-\infty<x<\infty$ , the constant $E$ (the total energy of the particle) must satisfy

errata.4 $E=E_{n}=\left(n+\tfrac{1}{2}\right)\hbar\omega,$ $n=0,1,2,\dots$ .

The corresponding eigenfunctions are

errata.5 $\eta_{n}(x)=\pi^{-\frac{1}{4}}2^{-\frac{1}{2}n}(n!\,b)^{-\frac{1}{2}}H_{n}% \left(x/b\right){\mathrm{e}}^{-x^{2}/2b^{2}},$

where $b=(\hbar/m\omega)^{1/2}$ , and $H_{n}$ is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

A second example is provided by the three-dimensional time-independent Schrödinger equation

errata.6 $\nabla^{2}\psi+\frac{2m}{\hbar^{2}}\left(E-V(\mathbf{x})\right)\psi=0,$

when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”
Section 18.40

The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

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Subsection 17.9(iii)

The title of the paragraph which was previously “Gasper’s $q$ -Analog of Clausen’s Formula” has been changed to “Gasper’s $q$ -Analog of Clausen’s Formula (16.12.2)”.

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Paragraph Inversion Formula (in §35.2)

The wording was changed to make the integration variable more apparent.

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Subsections 15.4(i), 15.4(ii)

Sentences were added specifying that some equations in these subsections require special care under certain circumstances. Also, (15.4.6) was expanded by adding the formula $F\left(a,b;a;z\right)=(1-z)^{-b}$ .

Report by Louis Klauder on 2017-01-01.

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9: Bibliography I

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M. E. H. Ismail and D. R. Masson (1994)

q

-Hermite polynomials, biorthogonal rational functions, and

q

-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:: ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.28(vii).
Encodings:: BibTeX

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A. R. Its and A. A. Kapaev (1987) The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

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Notes:: In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207
External Links:: ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

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A. R. Its and A. A. Kapaev (1998) Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.

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External Links:: ISSN 0305-4470, Document, MathReview (Yoshitsugu Takei), ZentralBlatt
Referenced by:: §32.11(v).
Encodings:: BibTeX

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K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida (1991) From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.

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External Links:: ISBN 3-528-06355-6, MathReview (Takeshi Sasaki), ZentralBlatt
Referenced by:: §32.2(vi), §32.7(vii).
Encodings:: BibTeX

10: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

… ►

Confluent Form

… ►For usage of the zeros of an OP in Gauss quadrature see §3.5(v). … ►

Degree lowering and raising differentiation formulas and structure relations

►For a large class of OP’s

p_{n}

there exist pairs of differentiation formulas …

Gauss–Hermite formula

1—10 of 11 matching pages

1: 3.5 Quadrature

Gauss–Hermite Formula

2: 18.5 Explicit Representations

§18.5(ii) Rodrigues Formulas

Hermite

Hermite

3: Bibliography G

4: 18.38 Mathematical Applications

Random Matrix Theory

5: 18.3 Definitions

§18.3 Definitions

6: 18.30 Associated OP’s

§18.30(iv) Associated Hermite Polynomials

7: Bibliography R

8: Errata

9: Bibliography I

10: 18.2 General Orthogonal Polynomials

§18.2(v) Christoffel–Darboux Formula

Confluent Form

Degree lowering and raising differentiation formulas and structure relations