Gauss–Laguerre formula

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8 matching pages

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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

… ►

§18.5(ii) Rodrigues Formulas

… ►Related formula: … ►

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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B. Gabutti and B. Minetti (1981) 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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L. Gatteschi (2002) 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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C. F. Gauss (1863) 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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W. Gautschi (2002b) 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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H. W. Gould (1972) 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

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

0<\nu<1

and

0<x<\infty

see Gautschi (2002a). …

6: Bibliography M

… ►

T. Masuda, Y. Ohta, and K. Kajiwara (2002) 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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C. Micu and E. Papp (2005) 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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S. C. Milne (1988) 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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D. S. Moak (1981) 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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D. S. Moak (1984) 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

… ►

N. M. Temme (1986) 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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N. M. Temme (1990a) 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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N. M. Temme (2003) 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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L. N. Trefethen (2008) 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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F. G. Tricomi (1949) 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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Changes

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In Paragraph Inversion Formula in §35.2, the wording was changed to make the integration variable more apparent.
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In many cases, the links from mathematical symbols to their definitions were corrected or improved. These links were also enhanced with ‘tooltip’ feedback, where supported by the user’s browser.

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Other Changes

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The overloaded operator $\equiv$ is now more clearly separated (and linked) to two distinct cases: equivalence by definition (in §§1.4(ii), 1.4(v), 2.7(i), 2.10(iv), 3.1(i), 3.1(iv), 4.18, 9.18(ii), 9.18(vi), 9.18(vi), 18.2(iv), 20.2(iii), 20.7(vi), 23.20(ii), 25.10(i), 26.15, 31.17(i)); and modular equivalence (in §§24.10(i), 24.10(ii), 24.10(iii), 24.10(iv), 24.15(iii), 24.19(ii), 26.14(i), 26.21, 27.2(i), 27.8, 27.9, 27.11, 27.12, 27.14(v), 27.14(vi), 27.15, 27.16, 27.19).
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The notation and markup for complex conjugation has been made more consistent in §§1.17(iii), 9.9(i), 10.11, 10.34, 10.63(ii), 12.11(ii), 13.7(ii), 14.30(ii), 23.5(iv), 28.12(ii), 31.15(iii), 34.3(vii), 36.2(iii), 36.2(iv), 36.8, 36.11.
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In Chapter 35 Functions of Matrix Argument, the generalized hypergeometric function of matrix argument ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ , was labeled inadvertently as its single variable counterpart ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ . Furthermore, the Jacobi function of matrix argument $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)$ , and the Laguerre function of matrix argument $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)$ , were also labeled inadvertently (and incorrectly) in terms of the single variable counterparts given by $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)$ , and $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)$ . In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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Other Changes

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To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

8.12.5 $\frac{{\mathrm{e}^{\pm\pi\mathrm{i}a}}}{2\mathrm{i}\sin(\pi a)}Q\left(-a,z{% \mathrm{e}^{\pm\pi\mathrm{i}}}\right)=\pm\tfrac{1}{2}\operatorname{erfc}\left(% \pm\mathrm{i}\eta\sqrt{a/2}\right)-\mathrm{i}T(a,\eta)$
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Following a suggestion from James McTavish on 2017-04-06, the recurrence relation $u_{k}=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}u_{k-1}$ was added to Equation (9.7.2).
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In §15.2(ii), the unnumbered equation

$\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}% \left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1% )!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right),$ $n=0,1,2,\dots$

was added in the second paragraph. An equation number will be assigned in an expanded numbering scheme that is under current development. Additionally, the discussion following (15.2.6) was expanded.
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In §15.4(i), due to a report by Louis Klauder on 2017-01-01, and in §15.4(iii), 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 $F\left(a,b;a;z\right)=(1-z)^{-b}$ .
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A bibliographic citation was added in §11.13(i).

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Other Changes

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In §1.2(i), a sentence was added after (1.2.1) to refer to (1.2.6) as the definition of the binomial coefficient $\genfrac{(}{)}{0.0pt}{}{z}{k}$ when $z$ is complex. As a notational clarification, wherever $n$ appeared originally in (1.2.6)–(1.2.9), it was replaced by $z$ .
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It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for $\operatorname{Ln}\Gamma\left(z+h\right)$ given in (5.11.8) is valid for $h$ $(\in\mathbb{C})$ ; originally it was unnecessarily restricted to $[0,1]$ .
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In §13.8(iii), a new paragraph with several new equations and a new reference has been added at the end to provide asymptotic expansions for Kummer functions $U\left(a,b,z\right)$ and ${\mathbf{M}}\left(a,b,z\right)$ as $a\to\infty$ in $|\operatorname{ph}a|\leq\pi-\delta$ and $b$ and $z$ fixed.
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Because of the use of the $O$ order symbol on the right-hand side, the asymptotic expansion (18.15.22) for the generalized Laguerre polynomial $L^{(\alpha)}_{n}\left(\nu x\right)$ was rewritten as an equality.
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The entire Section 27.20 was replaced.
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Bibliographic citations have been added or modified in §§2.4(v), 2.4(vi), 2.9(iii), 5.11(i), 5.11(ii), 5.17, 9.9(i), 10.22(v), 10.37, 11.6(iii), 11.9(iii), 12.9(i), 13.8(ii), 13.11, 14.15(i), 14.15(iii), 15.12(iii), 15.14, 16.11(ii), 16.13, 18.15(vi), 20.7(viii), 24.11, 24.16(i), 26.8(vii), 33.12(i), and 33.12(ii).
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Clarifications, typographic corrections, added or modified sentences appear in §§1.2(i), 1.10(i), 4.6(ii), 5.11(i), (11.11.1), (11.11.9), (21.5.7), and (27.14.7).

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Other Changes

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Equations (4.45.8) and (4.45.9) have been replaced with equations that are better for numerically computing $\operatorname{arctan}x$ .
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A new Subsection 13.29(v) Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.
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A new Subsection 14.5(vi) Addendum to §14.5(ii) $\mu=0$ , $\nu=2$ , containing the values of Legendre and Ferrers functions for degree $\nu=2$ has been added.
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Subsection 14.18(iii) has been altered to identify Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.
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A new Subsection 15.19(v) Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.
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Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.
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Bibliographic citations have been added in §§4.13, 5.6(i), 5.11(iii), 7.25(iii), 8.13(i), 10.37, 12.18, 14.11, 15.12(ii), 16.6, 18.16(ii), 18.16(iv), 18.24, 18.27(iv), 18.27(v),18.28(i), 24.13(i), 28.36(iii).
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Cross-references have been added in §§1.2(i), 10.19(iii), 10.23(ii), 17.2(iii), 18.15(iii), 19.2(iv), 19.16(i).
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Several small revisions have been made. For details see §§5.11(ii), 10.12, 10.19(ii), 18.9(i), 18.16(iv), 19.7(ii), 22.2, 32.11(v), 32.13(ii).
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Entries for the Sage computational system have been updated in the Software Index.
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The default document format for DLMF is now HTML5 which includes MathML providing better accessibility and display of mathematics.
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All interactive 3D graphics on the DLMF website have been recast using WebGL and X3DOM, improving portability and performance; WebGL it is now the default format.

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Gauss–Laguerre formula

8 matching pages

1: 3.5 Quadrature

Gauss–Laguerre Formula

2: 6.18 Methods of Computation

3: 18.5 Explicit Representations

§18.5(ii) Rodrigues Formulas

Laguerre

Laguerre

4: Bibliography G

5: 10.74 Methods of Computation

6: Bibliography M

7: Bibliography T

8: Errata