limit of Laguerre polynomials

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11: 18.19 Hahn Class: Definitions

§18.19 Hahn Class: Definitions

►The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator $\frac{d}{dx}$ in the case of the classical OP’s is played by a suitable difference operator. …In addition to the limit relations in §18.7(iii) there are limit relations involving the further families in the Askey scheme, see §§18.21(ii) and 18.26(ii). The Askey scheme, depicted in Figure 18.21.1, gives a graphical representation of these limits. … ►Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and standardization (§§18.2(i), 18.2(iii)) for the Hahn polynomials $Q_n(x;\alpha ,\beta ,N)$, Krawtchouk polynomials $K_n(x;p,N)$, Meixner polynomials $M_n(x;\beta ,c)$, and Charlier polynomials $C_n(x;a)$. …

12: 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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13: 18.10 Integral Representations

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Ultraspherical

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Legendre

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Laguerre

… ►for the Jacobi, Laguerre, and Hermite polynomials. … ►

Laguerre

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14: 1.17 Integral and Series Representations of the Dirac Delta

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1.17.4 $$\underset{n\to \mathrm{\infty }}{lim}{\delta }_n\left(x\right)=\delta \left(x\right),$$ $x\in ℝ$.

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), ${\delta }_n\left(x\right)$: Dirac delta sequence, $\in$: element of, $ℝ$: real line and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(i), §1.17 and Ch.1

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Legendre Polynomials (§§14.7(i) and 18.3)

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Laguerre Polynomials (§18.3)

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1.17.23 $$\delta \left(x-a\right)={\mathrm{e}}^{-(x+a)/2}\sum _{k=0}^{\mathrm{\infty }}{L}_k\left(x\right)L_k\left(a\right).$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\mathrm{e}$: base of natural logarithm, $L_n\left(x\right)=L_n^{(0)}\left(x\right)$: Laguerre polynomial and $k$: integer

Permalink:

http://dlmf.nist.gov/1.17.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

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Hermite Polynomials (§18.3)

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15: Bibliography D

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Strong and ratio asymptotics for Laguerre polynomials revisited. J. Math. Anal. Appl. 403 (2), pp. 477–486.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Pablo Manuel Román), ZentralBlatt

Referenced by:

§18.15(iv), §18.15(iv).

Encodings:

BibTeX

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On the real roots of Euler polynomials. Monatsh. Math. 106 (2), pp. 115–138.

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External Links:

ISSN 0026-9255, MathReview (H. M. Srivastava), ZentralBlatt

Referenced by:

§24.12(ii).

Encodings:

BibTeX

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On multiple zeros of Bernoulli polynomials. Acta Arith. 134 (2), pp. 149–155.

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External Links:

ISSN 0065-1036, Document, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.12(iv).

Encodings:

BibTeX

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Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (5), pp. 1029–1056.

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External Links:

ISSN 0036-1410, Document, MathReview (Kathi Karima Selig), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

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A Laguerre polynomial orthogonality and the hydrogen atom. Anal. Appl. (Singap.) 1 (2), pp. 177–188.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (E. Kreyszig), ZentralBlatt

Referenced by:

§18.39(ii), (33.14.15), Erratum (V1.0.24) for Equation (33.14.15).

Encodings:

BibTeX

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16: Bibliography F

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.

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External Links:

Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{VI}}$, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

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External Links:

ISSN 0027-7630, MathReview, ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v).

Encodings:

BibTeX

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Uniform asymptotic expansions of Laguerre polynomials. SIAM J. Math. Anal. 19 (5), pp. 1232–1248.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§18.15(iv).

Encodings:

BibTeX

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17: 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.