associated Laguerre functions

SearchAdvancedHelp

(0.003 seconds)

1—10 of 12 matching pages

1: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►The functions ${\varphi }_{n,\mathrm{\ell }}(r)$ defined by (33.14.14) are the hydrogenic bound states in attractive Coulomb potentials; their polynomial components are often called associated Laguerre functions; see Christy and Duck (1961) and Bethe and Salpeter (1977). …

2: 18.39 Applications in the Physical Sciences

… ►

d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s

… ►(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials ${𝐋}_{n+l}^{2l+1}({\rho }_n)$. …

3: 18.30 Associated OP’s

… ►

18.30.10 $${\int }_0^{\mathrm{\infty }}{L}_n^{\lambda }(x;c)L_m^{\lambda }(x;c)w^{\lambda }(x,c)dx=\frac{\mathrm{\Gamma }\left(n+c+\lambda +1\right)\mathrm{\Gamma }\left(c+1\right)}{{\left(c+1\right)}_n}{\delta }_{n,m},$$ $c+\lambda >-1$, $c\ge 0$, or $c+\lambda \ge 0$, $c>-1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $L_n^{\lambda }(x;c)$: associated Laguerre polynomial, $dx$: differential of $x$ , $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30(iii)

Permalink:

http://dlmf.nist.gov/18.30.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(iii), §18.30 and Ch.18

… ►

18.30.19 $$L_n^{\lambda }(x;c)=\underset{\varphi \to 0}{lim}{𝒫}_n^{(\lambda +1)/2}(\frac{-x}{2\sin \varphi };\varphi ,c),$$

ⓘ

Symbols:

$L_n^{\lambda }(x;c)$: associated Laguerre polynomial, ${𝒫}_n^{\lambda }(x;\varphi ,c)$: associated Meixner–Pollaczek polynomial, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Using the limit will convert (18.30.16) into (18.30.9). See also Askey and Wimp (1984, §2).

Permalink:

http://dlmf.nist.gov/18.30.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(v), §18.30 and Ch.18

…

4: Bibliography

… ►

Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

ⓘ

External Links:

MathReview (E. Weber), ZentralBlatt

Referenced by:

§33.9(ii).

Encodings:

BibTeX

… ►

SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.

ⓘ

Notes:

SPHEREPACK 2.0 is a collection of Fortran programs that facilitates computer modeling of geophysical processes. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms. The package also contains utility programs for computing the associated Legendre functions. SPHEREPACK 3.1 was released in August 2003.

External Links:

FullText, SPHEREPACK 3.1

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

ⓘ

External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

Encodings:

BibTeX

… ►

Associated Laguerre and Hermite polynomials. Proc. Roy. Soc. Edinburgh 96A, pp. 15–37.

ⓘ

External Links:

MathReview Entry

Referenced by:

(18.30.16), (18.30.17), (18.30.18), (18.30.19), (18.30.20), §18.30(iii), §18.30(iv), §18.30, §18.30(v), (18.35.10).

Encodings:

BibTeX

… ►

Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

ⓘ

External Links:

ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt

Referenced by:

§18.15(vi), §18.15(vi).

Encodings:

BibTeX

…

5: Errata

… ►

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, ${𝖯}_n,{𝖰}_n,P_n,Q_n,{𝑸}_n$ and the Laguerre polynomial, $L_n$, were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

…

6: Bibliography D

… ►

Strong and ratio asymptotics for Laguerre polynomials revisited. J. Math. Anal. Appl. 403 (2), pp. 477–486.

ⓘ

External Links:

ISSN 0022-247X, Document, FullText, MathReview (Pablo Manuel Román), ZentralBlatt

Referenced by:

§18.15(iv), §18.15(iv).

Encodings:

BibTeX

… ►

Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1), pp. 167–183.

ⓘ

External Links:

ISSN 0002-9947, Link, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

►

A Laguerre polynomial orthogonality and the hydrogen atom. Anal. Appl. (Singap.) 1 (2), pp. 177–188.

ⓘ

External Links:

ISSN 0219-5305, Document, Link, MathReview (E. Kreyszig), ZentralBlatt

Referenced by:

§18.39(ii), (33.14.15), Erratum (V1.0.24) for Equation (33.14.15).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

ⓘ

External Links:

ISSN 0308-2105, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.

Encodings:

BibTeX

►

Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.

ⓘ

External Links:

ISSN 0022-2526, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§14.15(iii), §2.8(i).

Encodings:

BibTeX

…

7: 10.74 Methods of Computation

… ► ►Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. … ►For applications of generalized Gauss–Laguerre quadrature (§3.5(v)) to the evaluation of the modified Bessel functions $K_{\nu }\left(z\right)$ for and see Gautschi (2002a). … ► ►

§10.74(vi) Zeros and Associated Values

…

8: 18.3 Definitions

§18.3 Definitions

►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►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). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

9: Bibliography L

… ►

Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

Encodings:

BibTeX

… ►

Orthogonal polynomials, duality and association schemes. SIAM J. Math. Anal. 13 (4), pp. 656–663.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (C. L. Parihar), ZentralBlatt

Referenced by:

§18.28(viii), §18.28(xi), §18.38(iii).

Encodings:

BibTeX

… ►

Co-recursive associated Jacobi polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 203–213.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

… ►

Polylogarithms and Associated Functions. North-Holland Publishing Co., New York.

ⓘ

Notes:

With a foreword by A. J. Van der Poorten

External Links:

ISBN 0-444-00550-1, MathReview, ZentralBlatt

Referenced by:

(25.12.10), (25.12.11), §25.12(ii), §25.12(i), §25.12(i), §25.12(ii), §25.12.

Encodings:

BibTeX

… ►

The spectral analysis of three families of exceptional Laguerre polynomials. J. Approx. Theory 202, pp. 5–41.

ⓘ

External Links:

ISSN 0021-9045, Document, Link, MathReview (Edmundo J. Huertas)

Referenced by:

§18.36(vi).

Encodings:

BibTeX

…

10: 18.1 Notation

… ►

Classical OP’s

… ►

Laguerre: $L_n^{(\alpha )}\left(x\right)$ and $L_n\left(x\right)=L_n^{(0)}\left(x\right)$. ($L_n^{(\alpha )}\left(x\right)$ with $\alpha \ne 0$ is also called Generalized Laguerre.)

… ►

$q$-Laguerre: $L_n^{(\alpha )}(x;q)$.

… ►

Associated OP’s

►

Associated OP’s are denoted via addition of the letter $c$ at the end of the listing of parameters in their usual notations.

…