Gauss–Legendre formula

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11: 18.12 Generating Functions

… ►The $z$-radii of convergence will depend on $x$, and in first instance we will assume $x\in [-1,1]$ for Jacobi, ultraspherical, Chebyshev and Legendre, $x\in [0,\mathrm{\infty })$ for Laguerre, and $x\in ℝ$ for Hermite. … ►and similar formulas as (18.12.3) and (18.12.3_5) by symmetry; compare the second row in Table 18.6.1. … ►

Legendre

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18.12.11 $$\frac{1}{\sqrt{1-2xz+z^2}}=\sum _{n=0}^{\mathrm{\infty }}{P}_n\left(x\right)z^n,$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.12

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.12 $${\mathrm{e}}^{xz}{J}_0\left(z\sqrt{1-x^2}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n\left(x\right)}{n!}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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12: Bibliography R

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Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.

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External Links:

ZentralBlatt

Referenced by:

§27.17.

Encodings:

BibTeX

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Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome I. Gauthier-Villars, Paris.

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Notes:

Préface de H. Villat

External Links:

MathReview (R. Campbell), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

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Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II. Gauthier-Villars, Paris.

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External Links:

MathReview (R. Campbell), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

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Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

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External Links:

ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

Referenced by:

§34.6.

Encodings:

BibTeX

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Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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External Links:

ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

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13: Bibliography C

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A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Gauss hypergeometric representations of the Ferrers function of the second kind. SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(14.13.2), §14.3(iii), §14.3(iii).

Encodings:

BibTeX

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An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.

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Notes:

Numerical integration and its complexity (Oberwolfach, 2001)

External Links:

ISSN 0885-064X, Document, Web Page, MathReview, ZentralBlatt

Referenced by:

§3.5(x).

Encodings:

BibTeX

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The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

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External Links:

ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.8(i).

Encodings:

BibTeX

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Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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External Links:

ISSN 0002-9920, MathReview (Willard Parker)

Referenced by:

§19.8(i).

Encodings:

BibTeX

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14: 2.10 Sums and Sequences

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§2.10(i) Euler–Maclaurin Formula

… ►This is the Euler–Maclaurin formula. Another version is the Abel–Plana formula: … ► … ►

Example

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15: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

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Confluent Form

… ►For usage of the zeros of an OP in Gauss quadrature see §3.5(v). … ►

Degree lowering and raising differentiation formulas and structure relations

►For a large class of OP’s $p_n$ there exist pairs of differentiation formulas …

16: 18.17 Integrals

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Legendre

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Legendre

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Legendre

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Legendre

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Legendre

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