Laguerre form

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1: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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Laguerre Form

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2: 18.11 Relations to Other Functions

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Laguerre

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3: 18.7 Interrelations and Limit Relations

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18.7.19 $$H_{2n}\left(x\right)={(-1)}^n{2}^{2n}n!L_n^{(-\frac{1}{2})}\left(x^2\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.40

Referenced by:

§18.16(v), §18.18(iii), §18.18(iv), §18.18(viii), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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18.7.20 $$H_{2n+1}\left(x\right)={(-1)}^n{2}^{2n+1}n!x{L}_n^{(\frac{1}{2})}\left(x^2\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.41

Referenced by:

§18.16(v), §18.18(iii), §18.18(iv), §18.7(ii), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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§18.7(iii) Limit Relations

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Jacobi $\to$ Laguerre

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Laguerre $\to$ Hermite

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4: 3.5 Quadrature

… ►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 …

5: 18.2 General Orthogonal Polynomials

… ►The generating functions (18.12.13), (18.12.15), (18.23.3), (18.23.4), (18.23.5) and (18.23.7) for Laguerre, Hermite, Krawtchouk, Meixner, Charlier and Meixner–Pollaczek polynomials, respectively, can be written in the form (18.2.45). …

6: 18.17 Integrals

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Laguerre

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Laguerre

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Laguerre

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Laguerre

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Laguerre

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7: 37.16 Orthogonal Polynomials on the Hyperoctant

… ►The OPs of degree $n$ with respect to the inner product (37.16.3) form the space ${𝒱}_n^d={𝒱}_n^{𝜶}({ℝ}_{+}^d)$. … ►Obviously, an orthogonal basis of ${𝒱}_n^{𝜶}({ℝ}_{+}^d)$ consisting of products of Laguerre polynomials is given by … ►Define Laguerre–Jacobi polynomials on the hyperoctant by …The polynomials $Q_{𝝂,k}^{𝜶}$ ($|𝝂|+k=n$) form a second orthogonal basis of ${𝒱}_n^{𝜶}({ℝ}_{+}^d)$. …

8: 18.15 Asymptotic Approximations

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§18.15(iv) Laguerre

… ►Here $J_{\nu }\left(z\right)$ denotes the Bessel function (§10.2(ii)), $\mathrm{env}{J}_{\nu }\left(z\right)$ denotes its envelope (§2.8(iv)), and $\delta$ is again an arbitrary small positive constant. … ►The asymptotic behavior of the classical OP’s as $x\to \pm \mathrm{\infty }$ with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. ►For asymptotic approximations of Jacobi, ultraspherical, and Laguerre polynomials in terms of Hermite polynomials, see López and Temme (1999a). These approximations apply when the parameters are large, namely $\alpha$ and $\beta$ (subject to restrictions) in the case of Jacobi polynomials, $\lambda$ in the case of ultraspherical polynomials, and $|\alpha |+|x|$ in the case of Laguerre polynomials. …

9: Bibliography T

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:

ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

Encodings:

BibTeX

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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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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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Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.

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External Links:

ISSN 0002-9947, Document, FullText, MathReview (John M. Voight), ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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10: 37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

… ►For the Laguerre polynomial $L_n^{(\alpha )}\left(x\right)$ and the Jacobi polynomial $P_n^{(\alpha ,\beta )}\left(x\right)$ in one variable see Table 18.3.1. Obviously, an orthogonal basis of ${𝒱}_n^{\alpha ,\beta }$ consisting of products of Laguerre polynomials is given by … ►Define Laguerre–Jacobi polynomials on the quarter plane by …These polynomials form a second orthogonal basis of ${𝒱}_n^{\alpha ,\beta }$, … ►In particular, for the orthogonal basis (37.5.4) of products of Laguerre polynomials, …