asymptotics of Laguerre polynomials

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11: Errata

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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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12: 18.34 Bessel Polynomials

§18.34 Bessel Polynomials

… ►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ and the generalized hypergeometric function ${}_2{}^{}F_0^{}$, the Laguerre polynomial $L_n^{(\alpha )}$ and the Whittaker function $W_{\kappa ,\mu }$ see §16.2(ii), §16.2(iv), (18.5.12), and (13.14.3), respectively. … ►expressed in terms of Romanovski–Bessel polynomials, Laguerre polynomials or Whittaker functions, we have …In this limit the finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ which is orthogonal on $(1,\mathrm{\infty })$ (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on $(0,\mathrm{\infty })$ (see (18.34.5_5)). … ►For uniform asymptotic expansions in terms of Hermite polynomials see López and Temme (1999b). …

13: Bibliography E

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Further results on McMahon’s asymptotic approximations. J. Phys. A 33 (36), pp. 6333–6341.

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External Links:

ISSN 0305-4470, Document, MathReview (R. G. Aĭrapetyan), ZentralBlatt

Referenced by:

§10.21(vi).

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Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§18.15(i).

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BibTeX

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Asymptotic Expansions. Dover Publications Inc., New York.

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External Links:

MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

Ch.2.

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The Sobolev orthogonality and spectral analysis of the Laguerre polynomials $\{L_n^{-k}\}$ for positive integers $k$ . J. Comput. Appl. Math. 171 (1-2), pp. 199–234.

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ISSN 0377-0427, Document, Link, MathReview (Khélifa Trimèche)

Referenced by:

§18.36(v).

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Note on the $X_1$-Laguerre orthogonal polynomials.

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Notes:

arXiv:0811.3559v2

Referenced by:

§18.36(vi).

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14: Bibliography L

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A uniform asymptotic expansion for Krawtchouk polynomials. J. Approx. Theory 106 (1), pp. 155–184.

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ISSN 0021-9045, Document, MathReview (Ahmed I. Zayed), ZentralBlatt

Referenced by:

§18.24.

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On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. Constr. Approx. 17 (1), pp. 59–90.

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ISSN 0176-4276, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§18.24.

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The spectral analysis of three families of exceptional Laguerre polynomials. J. Approx. Theory 202, pp. 5–41.

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ISSN 0021-9045, Document, Link, MathReview (Edmundo J. Huertas)

Referenced by:

§18.36(vi).

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Global asymptotics of the Hahn polynomials. Anal. Appl. (Singap.) 11 (3), pp. 1350018, 47.

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External Links:

ISSN 0219-5305, Document, FullText, MathReview (Yamilet Quintana), ZentralBlatt

Referenced by:

§18.24, §18.24.

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Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.

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External Links:

ISSN 0022-247X, Document, MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.34(iii), §24.11, §24.16(i), §24.16(i).

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15: Bibliography M

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Applying $q$-Laguerre polynomials to the derivation of $q$-deformed energies of oscillator and Coulomb systems. Romanian Reports in Physics 57 (1), pp. 25–34.

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Referenced by:

§17.17.

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Asymptotic approximations for prolate spheroidal wave functions. Studies in Appl. Math. 54 (4), pp. 315–349.

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External Links:

MathReview (M. L. Mehta), ZentralBlatt

Referenced by:

§30.9(i).

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Exceptional orthogonal polynomials.

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Notes:

Lecture at ICMAT School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, available at https://www.icmat.es/RT/optrim/school/lectures/Milson/icmat17.pdf

Referenced by:

§18.36(vi), §18.38(iii).

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.11, §29.7(ii).

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16: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(v) Asymptotic Solutions

… ►The functions ${\varphi }_{n,\mathrm{\ell }}(r)$ defined by (33.14.14) are the hydrogenic bound states in attractive Coulomb potentials; their polynomial components are often called associated Laguerre functions; see Christy and Duck (1961) and Bethe and Salpeter (1977). …

17: Bibliography K

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Uniform asymptotic approximations for the Meixner-Sobolev polynomials. Anal. Appl. (Singap.) 10 (3), pp. 345–361.

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ISSN 0219-5305, Document, FullText, MathReview (Diego E. Dominici), ZentralBlatt

Referenced by:

§18.24, §18.24.

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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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ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt

Referenced by:

§18.36(ii).

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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).

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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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ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

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Strong asymptotics of polynomials orthogonal with respect to Freud weights. Internat. Math. Res. Notices 1999 (6), pp. 299–333.

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External Links:

ISSN 1073-7928, Document, MathReview (Heinrich Begehr), ZentralBlatt

Referenced by:

§18.32.

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18: 18.2 General Orthogonal Polynomials

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Kernel Polynomials

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§18.2(vi) Zeros

… ►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). In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system) … ►Equations (18.14.3_5) and (18.14.8), both for $\alpha =0$, can be seen as special cases of this result for Jacobi and Laguerre polynomials, respectively.