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31: 18.3 Definitions

§18.3 Definitions

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As given by a Rodrigues formula (18.5.5).

►Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). … ►For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►For another version of the discrete orthogonality property of the polynomials $T_n\left(x\right)$ see (3.11.9). …

32: 3.8 Nonlinear Equations

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§3.8(iii) Other Methods

… ►For other efficient derivative-free methods, see Le (1985). … ►For the computation of zeros of orthogonal polynomials as eigenvalues of finite tridiagonal matrices (§3.5(vi)), see Gil et al. (2007a, pp. 205–207). … ►The polynomial … ►

Example. Wilkinson’s Polynomial

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33: Bibliography I

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On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:

ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

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An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.

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External Links:

ISSN 0030-8730, FullText, MathReview (T. Erber), ZentralBlatt

Referenced by:

§18.39(v).

Encodings:

BibTeX

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More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21 (1-2), pp. 191–204.

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External Links:

ISSN 0163-0563, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.39(v).

Encodings:

BibTeX

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Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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External Links:

ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt

Referenced by:

§1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

Encodings:

BibTeX

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34: About the Project

… ► Olver, Editor-in-Chief and Mathematics Editor of the DLMF, the other Editors initiated an effort aimed at updating the organizational structure of the DLMF project. … ►They were selected as recognized leaders in the research communities interested in the mathematics and applications of special functions and orthogonal polynomials; in the presentation of mathematics reference information online and in handbooks; and in the presentation of mathematics on the web. … ►They will be called upon to help deal with reports of suspected errors and suggestions for additions or other modifications to the chapters. … ►The complete list of Editors, Senior Associate Editors, Associate Editors, and other currently active contributors to the DLMF Project are listed on the Staff page. …

35: 18.41 Tables

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§18.41(i) Polynomials

►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_n\left(x\right)$, $U_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_n\left(x\right)$ and $U_n\left(x\right)$, and $0.5,1,3,5,10$ for $L_n\left(x\right)$ and $H_n\left(x\right)$. … ►

§18.41(iii) Other Tables

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36: 18.33 Polynomials Orthogonal on the Unit Circle

§18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(i) Definition

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§18.33(iii) Connection with OP’s on the Line

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§18.33(v) Biorthogonal Polynomials on the Unit Circle

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Recurrence Relations

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37: 29.20 Methods of Computation

… ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. … ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. … ►

§29.20(ii) Lamé Polynomials

… ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. … ►

§29.20(iii) Zeros

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38: 31.9 Orthogonality

§31.9 Orthogonality

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§31.9(i) Single Orthogonality

… ►The right-hand side may be evaluated at any convenient value, or limiting value, of $\zeta$ in $(0,1)$ since it is independent of $\zeta$. ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). ►

§31.9(ii) Double Orthogonality

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39: 18.10 Integral Representations

§18.10 Integral Representations

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Ultraspherical

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Legendre

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§18.10(iv) Other Integral Representations

… ►See also §18.17.

40: 18.18 Sums

§18.18 Sums

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§18.18(ii) Addition Theorems

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§18.18(iii) Multiplication Theorems

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§18.18(v) Linearization Formulas

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