difference operators

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31: 16.3 Derivatives and Contiguous Functions

… ►Other versions of these identities can be constructed with the aid of the operator identity … ►Two generalized hypergeometric functions ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ are (generalized) contiguous if they have the same pair of values of $p$ and $q$, and corresponding parameters differ by integers. …

32: 16.8 Differential Equations

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16.8.3 $$\left(\mathit{\vartheta }(\mathit{\vartheta }+b_1-1)\mathrm{\cdots }(\mathit{\vartheta }+b_q-1)-z(\mathit{\vartheta }+a_1)\mathrm{\cdots }(\mathit{\vartheta }+a_p)\right)w=0.$$

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Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $\mathit{\vartheta }$: differential operator and $w$: solution

Referenced by:

§16.8(ii), §16.8(ii), §16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E3

Encodings:

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See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.4 $$z^q{𝐷}^{q+1}w+\sum _{j=1}^q{z}^{j-1}({\alpha }_{j}z+{\beta }_j){𝐷}^{j}w+{\alpha }_{0}w=0,$$ $p\le q$,

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Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $𝐷$: differential operator, $w$: solution, ${\alpha }_j$: constant and ${\beta }_j$: constant

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E4

Encodings:

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See also:

Annotations for §16.8(ii), §16.8 and Ch.16

… ►When no $b_j$ is an integer, and no two $b_j$ differ by an integer, a fundamental set of solutions of (16.8.3) is given by … ►When $p=q+1$, and no two $a_j$ differ by an integer, another fundamental set of solutions of (16.8.3) is given by … ►When $p=q+1$ and some of the $a_j$ differ by an integer a limiting process can again be applied. …

33: Bibliography J

… ►

Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.

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Notes:

Third edition published in 1965 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI

External Links:

MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§26.1, §26.1, §26.8(vii), §5.16, Notations S.

Encodings:

BibTeX

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Calculus of Finite Differences. 3rd edition, AMS Chelsea, Providence, RI.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§24.17(i), §24.6.

Encodings:

BibTeX

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Operator Techniques in Atomic Spectroscopy. Princeton University Press, Princeton, NJ.

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External Links:

ISBN 0-691-05901-2

Referenced by:

§34.12, Profile Brian R. Judd.

Encodings:

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34: 31.17 Physical Applications

… ►We use vector notation $[𝐬,𝐭,𝐮]$ (respective scalar $(s,t,u)$) for any one of the three spin operators (respective spin values). … ►

$${𝐻}_s\mathrm{\Psi }(𝐱)\equiv (-2𝐬\cdot 𝐭-(2/a)𝐬\cdot 𝐮)\mathrm{\Psi }(𝐱)=h_s\mathrm{\Psi }(𝐱),$$

… ►The operators ${𝐉}^2$ and ${𝐻}_s$ admit separation of variables in $z_1,z_2$, leading to the following factorization of the eigenfunction $\mathrm{\Psi }(𝐱)$: … ►For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …

35: 2.3 Integrals of a Real Variable

… ►is finite and bounded for $n=0,1,2,\mathrm{\dots }$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^n(x-{\sigma }_n))|$ when $x$ exceeds both $0$ and ${\sigma }_n$. … ►In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by ►

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

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Symbols:

$dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $a$: left endpoint and $b$: right endpoint

Referenced by:

§10.17(iv), Erratum (V1.1.1) for Equation (2.3.6)

Permalink:

http://dlmf.nist.gov/2.3.E6

Encodings:

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Correction (effective with 1.1.1):

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

Suggested 2021-02-08 by Juan Luis Varona

See also:

Annotations for §2.3(i), §2.3 and Ch.2

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36: 18.36 Miscellaneous Polynomials

… ►These are polynomials in one variable that are orthogonal with respect to a number of different measures. … ►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the $L_n^{(-k)}(x)$ polynomials, self-adjointness implying both orthogonality and completeness. … ►Completeness and orthogonality follow from the self-adjointness of the corresponding Schrödinger operator, Gómez-Ullate and Milson (2014), Marquette and Quesne (2013).

37: DLMF Project News

error generating summary

38: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►Analogous to (18.39.7) the 3D Schrödinger operator is …where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►Here tridiagonal representations of simple Schrödinger operators play a similar role. The radial operator (18.39.28) …

39: 14.30 Spherical and Spheroidal Harmonics

… ►Sometimes $Y_{l,m}(\theta ,\varphi )$ is denoted by ${\mathrm{i}}^{-l}{𝔇}_{lm}(\theta ,\varphi )$; also the definition of $Y_{l,m}(\theta ,\varphi )$ can differ from (14.30.1), for example, by inclusion of a factor ${(-1)}^m$. … ►

Parity Operation

… ►Here, in spherical coordinates, $L^2$ is the squared angular momentum operator: …and ${\mathrm{L}}_z$ is the $z$ component of the angular momentum operator ►

14.30.13 $${\mathrm{L}}_z=-\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial \varphi };$$

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Defines:

${\mathrm{L}}_z$: $z$ component of the angular momentum operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: complex variable

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E13

Encodings:

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Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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40: Bibliography L

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Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (2), pp. 425–452.

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Notes:

For a Maple implementation of the algorithm given in this paper for Jack and zonal polynomials see Zeilberger (website).

External Links:

ISSN 0010-3616, Link, MathReview (Kazuhiro Hikami), ZentralBlatt

Referenced by:

§35.12.

Encodings:

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Introduction to spectral theory: selfadjoint ordinary differential operators. Translations of Mathematical Monographs, Vol. 39, American Mathematical Society, Providence, R.I..

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Notes:

Translated from the Russian by Amiel Feinstein

External Links:

MathReview Entry

Referenced by:

§1.18(x).

Encodings:

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Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

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External Links:

ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

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Numerical Solution of Linear Difference Equations. NBSIR Technical Report 80-1976, National Bureau of Standards, Gaithersburg, MD 20899.

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Notes:

(Available from National Technical Information Service, Springfield, VA 22161.)

Referenced by:

§16.25, §3.6(vii).

Encodings:

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The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

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Notes:

In Russian. English translation: Differential Equations 7(6), pp. 853–854

External Links:

MathReview (Z. Rubinstein), ZentralBlatt

Referenced by:

§32.7(ii).

Encodings:

BibTeX

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