Laguerre form

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21: 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. … ►

Laguerre

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18.12.13 $${(1-z)}^{-\alpha -1}\exp \left(\frac{xz}{z-1}\right)=\sum _{n=0}^{\mathrm{\infty }}{L}_n^{(\alpha )}\left(x\right)z^n,$$ .

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\exp z$: exponential function, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.15

Referenced by:

§18.12, §18.18(ii), §18.18(iv), §18.18(viii), §18.2(xii), (8.7.6)

Permalink:

http://dlmf.nist.gov/18.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.14 $$\mathrm{\Gamma }\left(\alpha +1\right){(xz)}^{-\frac{1}{2}\alpha }{\mathrm{e}}^z{J}_{\alpha }\left(2\sqrt{xz}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(\alpha )}\left(x\right)}{{\left(\alpha +1\right)}_n}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.16

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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22: Bibliography

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Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

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Notes:

Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.

External Links:

Document, MathReview (G. R. Belickii), ZentralBlatt

Referenced by:

§36.2(i).

Encodings:

BibTeX

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Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 30(1975), no. 5, pp. 1–75.

External Links:

Document, MathReview (D. Siersma), ZentralBlatt

Referenced by:

§36.2(i), Ch.36.

Encodings:

BibTeX

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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:

BibTeX

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Commentary on the Paper “Beiträge zur Theorie der Toeplitzschen Form”. In Gábor Szegő, Collected Papers. Vol. 1, Contemporary Mathematicians, pp. 303–305.

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External Links:

ISBN 3-7643-3056-2, MathReview (G. Piranian)

Referenced by:

§18.33(iv), §18.33(iv), §18.33(v).

Encodings:

BibTeX

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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt

Referenced by:

§18.15(vi), §18.15(vi).

Encodings:

BibTeX

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23: Bibliography K

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Closed-form representations of the Lambert $W$ function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.

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External Links:

ISSN 1311-0454, MathReview (Lewis H. Ryder), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

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A vortex filament moving without change of form. J. Fluid Mech. 112, pp. 397–409.

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External Links:

ISSN 0022-1120, Document, MathReview, ZentralBlatt

Referenced by:

§19.35(ii).

Encodings:

BibTeX

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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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