integral representation of Laguerre polynomials

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11: 18.27 $q$-Hahn Class

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Little $q$-Laguerre polynomials

►Little $q$-Jacobi polynomials $p_n(x;a,b;q)$ for $b=0$ are called little $q$-Laguerre or Wall polynomials: … ►

From Little $q$-Laguerre to Laguerre

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§18.27(v) $q$-Laguerre Polynomials

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From $q$-Laguerre to Laguerre

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12: Errata

… ►We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. … ►

8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$,

The constraint was updated to include “”.

Suggested by Walter Gautschi on 2022-10-14

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The generalized hypergeometric function of matrix argument ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$, was linked inadvertently as its single variable counterpart ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$. Furthermore, the Jacobi function of matrix argument $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and the Laguerre function of matrix argument $L_{\nu }^{(\gamma )}\left(𝐓\right)$, were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and $L_{\nu }^{(\gamma )}\left(𝐓\right)$. In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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Because of the use of the $O$ order symbol on the right-hand side, the asymptotic expansion for the generalized Laguerre polynomial $L_n^{(\alpha )}\left(\nu x\right)$ was rewritten as an equality.

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13: 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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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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Une nouvelle représentation explicite des nombres d’Euler. C. R. Acad. Sci. Paris Sér. A-B 286 (19), pp. A807–A809.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§24.6.

Encodings:

BibTeX

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Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.

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Notes:

Papers from the conference held in honor of Ray A. Kunze at the AMS Special Session, Cincinnati, Ohio, January 12–14, 1994

External Links:

ISBN 0-8218-0310-7, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Donald St. P. Richards.

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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14: 18.39 Applications in the Physical Sciences

… ►The functions ${\varphi }_n$ are expressed in terms of Romanovski–Bessel polynomials, or Laguerre polynomials by (18.34.7_1). … ► $p$ here being the order of the Laguerre polynomial, $L_p^{(2l+1)}$ of Table 18.8.1, line 11, and $l$ the angular momentum quantum number, and where … ►The associated Coulomb–Laguerre polynomials are defined as … ►As the scattering eigenfunctions of Chapter 33, are not OP’s, their further discussion is deferred to §18.39(iv), where discretized representations of these scattering states are introduced, Laguerre and Pollaczek OP’s then playing a key role. … ►For physical applications of $q$-Laguerre polynomials see §17.17. …