Poisson kernel for Laguerre polynomials

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1: 18.18 Sums

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Laguerre

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2: 37.16 Orthogonal Polynomials on the Hyperoctant

… ►Define Laguerre–Jacobi polynomials on the hyperoctant by … ►

§37.16(ii) Poisson Kernel

►The Poisson kernel (37.13.6) of ${𝒱}_n^{𝜶}({ℝ}_{+}^d)$ is given explicitly by … …

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error generating summary

4: 18.12 Generating Functions

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Jacobi

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Ultraspherical

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Legendre

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Laguerre

… ►See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.

5: 37.17 Hermite Polynomials on ${ℝ}^d$

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§37.17(i) Product Hermite Polynomials

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§37.17(ii) Spherical Hermite Polynomials

… ►The polynomials $S_{Y,k,n}^{\alpha }$ are called spherical Hermite polynomials. … ►The Poisson kernel (37.13.6) of ${𝒱}_n({ℝ}^d)$ is given explicitly by the Mehler formula … ►These polynomials are introduced in (Appell and Kampé de Fériet, 1926). …

6: 18.2 General Orthogonal Polynomials

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Kernel Polynomials

… ►Then the kernel polynomials … ►

Poisson kernel

►For OP’s $p_n$ with $h_n$ and orthogonality relation as in (18.2.5) and (18.2.5_5), the Poisson kernel is defined by …Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12). …

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).

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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).

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Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

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On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes. Phys. A 244 (1-4), pp. 402–413.

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ISSN 0378-4371, Document, MathReview (Vitor R. Vieira)

Referenced by:

§32.11(iv).

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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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