q-Laguerre polynomials

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31: 24.13 Integrals

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§24.13(i) Bernoulli Polynomials

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24.13.3 $${\int }_x^{x+(1/2)}{B}_n\left(t\right)dt=\frac{E_n\left(2x\right)}{2^{n+1}},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer, $t$: real or complex and $x$: real or complex

Referenced by:

§24.13(i)

Permalink:

http://dlmf.nist.gov/24.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(i), §24.13 and Ch.24

… ►For integrals of the form ${\int }_0^x{B}_n\left(t\right)B_m\left(t\right)dt$ and ${\int }_0^x{B}_n\left(t\right)B_m\left(t\right)B_k\left(t\right)dt$ see Agoh and Dilcher (2011). ►

§24.13(ii) Euler Polynomials

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§24.13(iii) Compendia

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32: 16.26 Approximations

… ►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

33: 18.38 Mathematical Applications

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Approximation Theory

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Integrable Systems

… ►Ultraspherical polynomials are zonal spherical harmonics. … ►

Group Representations

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Zhedanov Algebra

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34: 29.16 Asymptotic Expansions

§29.16 Asymptotic Expansions

►Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. The approximations for Lamé polynomials hold uniformly on the rectangle $0\le \mathrm{\Re }z\le K$, $0\le \mathrm{\Im }z\le K^{\prime }$, when $nk$ and $n{k}^{\prime }$ assume large real values. …

35: 18.30 Associated OP’s

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§18.30(ii) Associated Legendre Polynomials

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§18.30(vi) Corecursive Orthogonal Polynomials

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Numerator and Denominator Polynomials

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§18.30(viii) Other Associated Polynomials

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36: 18.19 Hahn Class: Definitions

§18.19 Hahn Class: Definitions

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Hahn, Krawtchouk, Meixner, and Charlier

►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 $C_n(x;a)$. … ► … ► …

37: 18 Orthogonal Polynomials

Chapter 18 Orthogonal Polynomials

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38: Tom H. Koornwinder

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. … ►Koornwinder has been active as an officer in the SIAM Activity Group on Special Functions and Orthogonal Polynomials. … ►

18 Orthogonal Polynomials

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39: 18.11 Relations to Other Functions

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Ultraspherical

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Laguerre

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Hermite

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Jacobi

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Laguerre

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40: 29.12 Definitions

§29.12 Definitions

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§29.12(i) Elliptic-Function Form

… ►There are eight types of Lamé polynomials, defined as follows: …In consequence they are doubly-periodic meromorphic functions of $z$. … ►

§29.12(ii) Algebraic Form

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