Gauss–Laguerre formula

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1: 3.5 Quadrature

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Gauss–Laguerre Formula

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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 3.5.8: Nodes and weights for the 15-point Gauss–Laguerre formula.

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$x_k$		$w_k$
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Table 3.5.9: Nodes and weights for the 20-point Gauss–Laguerre formula.

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$x_k$		$w_k$
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2: 6.18 Methods of Computation

… ►For example, the Gauss-Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). …

3: 18.5 Explicit Representations

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§18.5(ii) Rodrigues Formulas

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Laguerre

… ►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). … ►

Laguerre

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4: 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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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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Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

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External Links:

ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§13.29(iii), §15.19(iii).

Encodings:

BibTeX

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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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5: 10.74 Methods of Computation

… ►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … ►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel 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). …

6: Bibliography M

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A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(v).

Encodings:

BibTeX

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Applying $q$-Laguerre polynomials to the derivation of $q$-deformed energies of oscillator and Coulomb systems. Romanian Reports in Physics 57 (1), pp. 25–34.

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Referenced by:

§17.17.

Encodings:

BibTeX

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey)

Referenced by:

§18.27(v).

Encodings:

BibTeX

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The $q$-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.

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External Links:

ISSN 0035-7596, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§5.18(ii).

Encodings:

BibTeX

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7: 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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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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Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.

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External Links:

ISSN 0036-1445, Document, MathReview (Kai Diethelm), ZentralBlatt

Referenced by:

§3.5(iv).

Encodings:

BibTeX

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Sul comportamento asintotico dell’$n$-esimo polinomio di Laguerre nell’intorno dell’ascissa $4n$ . Comment. Math. Helv. 22, pp. 150–167.

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External Links:

ISSN 0010-2571, Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§18.16(iv).

Encodings:

BibTeX

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8: Errata

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

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Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

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The generalized hypergeometric function of matrix argument ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$, was linked inadvertently as its single variable counterpart ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$. Furthermore, the Jacobi function of matrix argument $P_{\nu }^{(\gamma ,\delta )}\left(\mathbf{T}\right)$, and the Laguerre function of matrix argument $L_{\nu }^{(\gamma )}\left(\mathbf{T}\right)$, were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P_{\nu }^{(\gamma ,\delta )}\left(\mathbf{T}\right)$, and $L_{\nu }^{(\gamma )}\left(\mathbf{T}\right)$. In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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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 $F(a,b;a;z)={(1-z)}^{-b}$.

Report by Louis Klauder on 2017-01-01.

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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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