Laguerre polynomials

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31: 3.5 Quadrature

… ►The $p_n(x)$ are the monic Laguerre polynomials $L_n\left(x\right)$ (§18.3). ►

Table 3.5.6: Nodes and weights for the 5-point Gauss–Laguerre formula.

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$x_k$		$w_k$
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Table 3.5.7: Nodes and weights for the 10-point Gauss–Laguerre formula.

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$x_k$		$w_k$
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Table 3.5.17_5: Recurrence coefficients in (3.5.30) and (3.5.30_5) for monic versions $p_n(x)$ and orthonormal versions $q_n(x)$ of the classical orthogonal polynomials.

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$p_n(x)$	$q_n(x)$	${\alpha }_n$	${\beta }_n$	$h_0$
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$\frac{1}{k_n}{L}_n^{(\alpha )}\left(x\right)$	$\frac{{(-1)}^n}{\sqrt{h_n}}{L}_n^{(\alpha )}\left(x\right)$	$2n+\alpha +1$	$n(n+\alpha )$	$\mathrm{\Gamma }(\alpha +1)$
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►For the choice $q_n(x)=\frac{1}{\sqrt{h_n}}{L}_n^{(\alpha )}\left(x\right)$ the recurrence relation (3.5.30_5) takes the form …

32: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

… ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).

33: Bibliography Y

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Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.

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External Links:

MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

§35.10.

Encodings:

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34: 18.24 Hahn Class: Asymptotic Approximations

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Approximations in Terms of Laguerre Polynomials

… ►These approximations are in terms of Laguerre polynomials and hold uniformly for $\mathrm{ph}\left(x+i\lambda \right)\in [0,\pi ]$. …

35: Errata

… ►We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. … ►

18.34.1 $$y_n(x;a)={}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-n,n+a-1}{-};-\frac{x}{2})={\left(n+a-1\right)}_n{\left(\frac{x}{2}\right)}^n{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n}{-2n-a+2};\frac{2}{x})=n!{\left(-\frac{1}{2}x\right)}^n{L}_n^{(1-a-2n)}\left(2x^{-1}\right)={\left(\frac{1}{2}x\right)}^{1-\frac{1}{2}a}{\mathrm{e}}^{1/x}{W}_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right)$$

This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluent hypergeometric function.

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8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$,

The constraint was updated to include “”.

Suggested by Walter Gautschi on 2022-10-14

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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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Because of the use of the $O$ order symbol on the right-hand side, the asymptotic expansion for the generalized Laguerre polynomial $L_n^{(\alpha )}\left(\nu x\right)$ was rewritten as an equality.

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36: Bibliography T

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Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

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External Links:

FullText

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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Asymptotic estimates for Laguerre polynomials. Z. Angew. Math. Phys. 41 (1), pp. 114–126.

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External Links:

ISSN 0044-2275, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§18.11(i), §18.16(vi).

Encodings:

BibTeX

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37: Bibliography E

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The Sobolev orthogonality and spectral analysis of the Laguerre polynomials $\{L_n^{-k}\}$ for positive integers $k$ . J. Comput. Appl. Math. 171 (1-2), pp. 199–234.

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External Links:

ISSN 0377-0427, Document, Link, MathReview (Khélifa Trimèche)

Referenced by:

§18.36(v).

Encodings:

BibTeX

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Note on the $X_1$-Laguerre orthogonal polynomials.

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Notes:

arXiv:0811.3559v2

Referenced by:

§18.36(vi).

Encodings:

BibTeX

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38: Bibliography K

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On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

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External Links:

ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

(18.14.8), §18.14(i), §18.17(ii), §18.18(ii).

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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39: 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:

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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:

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40: Bibliography G

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A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Asymptotics and bounds for the zeros of Laguerre polynomials: A survey. J. Comput. Appl. Math. 144 (1-2), pp. 7–27.

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External Links:

ISSN 0377-0427, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.16(iv), §18.16(iv), §18.16(vi).

Encodings:

BibTeX

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