general orthogonal polynomials

SearchAdvancedHelp

(0.014 seconds)

21—30 of 74 matching pages

21: 18.33 Polynomials Orthogonal on the Unit Circle

… ►This states that for any sequence ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}$ with ${\alpha }_n\in ℂ$ and the polynomials ${\mathrm{\Phi }}_n(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with ${\mathrm{\Phi }}_0(z)=1$ satisfy the orthogonality relation (18.33.17) for a unique probability measure $\mu$ with infinite support on the unit circle. …

22: 18.36 Miscellaneous Polynomials

… ►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the $L_n^{(-k)}(x)$ polynomials, self-adjointness implying both orthogonality and completeness. …

23: 18.3 Definitions

§18.3 Definitions

… ►

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). … ►However, in general they are not orthogonal with respect to a positive measure, but a finite system has such an orthogonality. …

24: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

►

§18.21(i) Dualities

… ►

§18.21(ii) Limit Relations and Special Cases

… ►

Hahn $\to$ Jacobi

… ►

Meixner $\to$ Laguerre

…

25: 18.38 Mathematical Applications

… ►

Quadrature

… ►

Riemann–Hilbert Problems

… ►

Radon Transform

… ►

Group Representations

… ►

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

…

26: Bibliography Z

… ►

Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications. Dover, New York.

ⓘ

Notes:

Reprint of 1965 McGraw-Hill Publication

External Links:

ISBN 0-486-65479-6, MathReview Entry

Referenced by:

§1.16(ix).

Encodings:

BibTeX

►

Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

ⓘ

External Links:

ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

… ►

“Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.

ⓘ

External Links:

MathReview (Vadim B. Kuznetsov), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

►

On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.

ⓘ

External Links:

ISSN 0021-9045, Link, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.33(iii).

Encodings:

BibTeX

►

Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.

ⓘ

Referenced by:

§35.7(ii).

Encodings:

BibTeX

…

27: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
4.48(iii) General Precision	✓					✓	✓	✓				✓	✓								✓	✓	a		REDUCE
…
16 Generalized Hypergeometric Functions & Meijer G-Function
…
18 Orthogonal Polynomials
…
24 Bernoulli and Euler Polynomials
24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, $E_n\left(x\right)$	✓	✓						✓		✓		a	✓	✓			✓		✓		✓	✓	✓		Derive, MuPAD
…

…

28: 18.18 Sums

§18.18 Sums

… ►

§18.18(ii) Addition Theorems

… ►

§18.18(iii) Multiplication Theorems

… ►

§18.18(v) Linearization Formulas

… ►

29: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

►

§18.7(i) Linear Transformations

… ►

Legendre, Ultraspherical, and Jacobi

… ►

§18.7(ii) Quadratic Transformations

… ►

§18.7(iii) Limit Relations

…

30: 2.9 Difference Equations

… ►For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B). ►For applications of asymptotic methods for difference equations to orthogonal polynomials, see, e. …These methods are particularly useful when the weight function associated with the orthogonal polynomials is not unique or not even known; see, e. …