associated Laguerre polynomials

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1: 18.30 Associated OP’s

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§18.30(iii) Associated Laguerre Polynomials

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

$L_{-1}^{\lambda }(x;c)=0,$

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$L_0^{\lambda }(x;c)=1,$

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18.30.9 $$(n+c+1)L_{n+1}^{\lambda }(x;c)=(2n+2c+\lambda +1-x)L_n^{\lambda }(x;c)-(n+c+\lambda )L_{n-1}^{\lambda }(x;c),$$ $n=0,1,\mathrm{\dots }$.

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Symbols:

$L_n^{\lambda }(x;c)$: associated Laguerre polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.30.19)

Permalink:

http://dlmf.nist.gov/18.30.E9

Encodings:

pMML, png, TeX

See also:

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

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2: 18.39 Applications in the Physical Sciences

… ►The same solutions as in paragraph c), above, appear frequently in the literature in terms of associated Laguerre polynomials, which are referred to here as associated Coulomb–Laguerre polynomials to avoid confusion with the more recent meaning of ‘associated’ of §18.30. The associated Coulomb–Laguerre polynomials are defined as … ►(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 ${𝐋}_{n+l}^{2l+1}({\rho }_n)$. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …

3: Bibliography

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

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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:

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4: 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). …

5: 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. … ►

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

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6: Bibliography D

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

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External Links:

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

Referenced by:

§18.15(iv), §18.15(iv).

Encodings:

BibTeX

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Chebyshev expansion of the associated Legendre polynomial $P_L^M(x)$ . Comput. Phys. Comm. 18 (1), pp. 63–71.

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Notes:

Double-precision Fortran, maximum accuracy 25D

External Links:

Document, Code

Referenced by:

3rd item.

Encodings:

BibTeX

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

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External Links:

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

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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

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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

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

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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

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7: 37.18 Orthogonal Polynomials on Quadratic Domains

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§37.18(iii) Laguerre Polynomials on the Cone

►These are OPs on the unbounded cone ${𝕍}^{d+1}=\{(𝐱,t)\mid \Vert 𝐱\Vert \le t,t\in {ℝ}_{+},𝐱\in {ℝ}^d\}$ associated to the Laguerre weight function …The OPs in the basis (37.18.3) are given in terms of the Laguerre polynomials and OPs in ${𝒱}_m({𝔹}^d)$ with respect to $W_{\mu -\frac{1}{2}}$: … ► … ► …

8: 37.12 Orthogonal Polynomials on Quadratic Surfaces

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§37.12(iii) Laguerre Polynomials on the Conic Surface

►These are OPs on the unbounded conic surface ${𝕍}_0^{d+1}=\{(𝐱,t)\mid \Vert 𝐱\Vert =t,t\in {ℝ}_{+},𝐱\in {ℝ}^d\}$ associated to the Laguerre weight …The OPs in the basis (37.12.2) are given in terms of Laguerre polynomials: … ► … ► …

9: 18.1 Notation

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Classical OP’s

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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.)

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

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Associated OP’s

… ►Nor do we consider the shifted Jacobi polynomials: …

10: 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)). …