Gaussian polynomials

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11: 18.38 Mathematical Applications

… ►

Approximation Theory

… ►Classical OP’s play a fundamental role in Gaussian quadrature. … ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►

Integrable Systems

… ►Ultraspherical polynomials are zonal spherical harmonics. …

12: 35.10 Methods of Computation

… ►For small values of

\|\mathbf{T}\|

the zonal polynomial expansion given by (35.8.1) can be summed numerically. … ►See Yan (1992) for the

{{}_{1}F_{1}}

and

{{}_{2}F_{1}}

functions of matrix argument in the case

m=2

, and Bingham et al. (1992) for Monte Carlo simulation on

\mathbf{O}(m)

applied to a generalization of the integral (35.5.8). ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

13: 3.5 Quadrature

… ►For effective testing of Gaussian quadrature rules see Gautschi (1983). … ►

Gauss–Legendre Formula

… ►The

p_{n}(x)

are the monic Hermite polynomials

H_{n}\left(x\right)

(§18.3). … ►Oscillatory integral transforms are treated in Wong (1982) by a method based on Gaussian quadrature. … ► …

14: 18.36 Miscellaneous Polynomials

… ►EOP’s are non-classical in that not only are certain polynomial orders missing, but, also, not all EOP polynomial zeros are within the integration range of their generating measure, and EOP-orthogonality properties do not allow development of Gaussian-type quadratures. …

15: Bibliography R

… ►

M. Rahman (1981) A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:: ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

►

M. Rahman (2001) The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.

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External Links:: MathReview (Wadim Zudilin), ZentralBlatt
Referenced by:: §18.30(viii).
Encodings:: BibTeX

… ►

W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:: arXiv:1812.02196
Referenced by:: §18.40(ii).
Encodings:: BibTeX

… ►

W. P. Reinhardt (2021b) Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:: ISSN 1521-9615, Document, Link
Referenced by:: §18.39(iv), §18.40(ii).
Encodings:: BibTeX

… ►

D. St. P. Richards (Ed.) (1992) Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.

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External Links:: ISBN 0-8218-5159-4, MathReview, ZentralBlatt
Referenced by:: Ch.35, Profile Donald St. P. Richards.
Encodings:: BibTeX

…

16: Bibliography F

… ►

J. L. Fields and Y. L. Luke (1963a) 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

►

J. L. Fields and Y. L. Luke (1963b) 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

… ►

J. L. Fields (1965) 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

… ►

B. D. Fried and S. D. Conte (1961) The Plasma Dispersion Function: The Hilbert Transform of the Gaussian. Academic Press, London-New York.

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Notes:: Erratum: Math. Comp. v. 26 (1972), no. 119, p. 814.
External Links:: MathReview (R. Balescu)
Referenced by:: §7.21.
Encodings:: BibTeX

… ►

Y. V. Fyodorov (2005) Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §18.38(ii).
Encodings:: BibTeX

17: Bibliography G

… ►

G. Gasper (1972) An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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External Links:: ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt
Referenced by:: §18.14(ii).
Encodings:: BibTeX

… ►

W. Gautschi (1983) How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.

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External Links:: ISSN 0006-3835, Document, MathReview (G. A. Evans), ZentralBlatt
Referenced by:: §18.40(ii), §3.5(v), §9.17(iii).
Encodings:: BibTeX

►

W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §3.8(vi).
Encodings:: BibTeX

… ►

W. Gautschi (2002a) Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.

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External Links:: ISSN 0006-3835, Document, MathReview (Khélifa Trimèche), ZentralBlatt
Referenced by:: §10.74(iii), §9.17(iii).
Encodings:: BibTeX

… ►

V. X. Genest, L. Vinet, and A. Zhedanov (2016) The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.

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External Links:: ISSN 0002-9939, Link, MathReview (Raj Wilson), ZentralBlatt
Referenced by:: §18.28(xi).
Encodings:: BibTeX

…

18: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.1

{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},

-c+\frac{1}{2}(j+1)\notin\mathbb{N}

1\leq j\leq m

;

\|\mathbf{T}\|<1

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\|\mathbf{T}\|$ : spectral norm of $\mathbf{T}$ , $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(i), §35.7 and Ch.35

…

19: Bibliography H

… ►

B. A. Hargrave and B. D. Sleeman (1977) Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

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External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.16.
Encodings:: BibTeX

… ►

E. Hendriksen and H. van Rossum (1986) Orthogonal Laurent polynomials. Nederl. Akad. Wetensch. Indag. Math. 48 (1), pp. 17–36.

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External Links:: ISSN 0019-3577, Document, MathReview (W. J. Thron), ZentralBlatt
Referenced by:: §18.33(v).
Encodings:: BibTeX

… ►

F. T. Howard (1976) Roots of the Euler polynomials. Pacific J. Math. 64 (1), pp. 181–191.

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External Links:: MathReview (M. Marden), ZentralBlatt
Referenced by:: §24.12(ii).
Encodings:: BibTeX

… ►

Y. P. Hsu (1993) Development of a Gaussian hypergeometric function code in complex domains. Internat. J. Modern Phys. C 4 (4), pp. 805–840.

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External Links:: ISSN 0129-1831, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §15.20(iii).
Encodings:: BibTeX

… ►

I. Huang and S. Huang (1999) Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.

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External Links:: ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

…

20: 18.39 Applications in the Physical Sciences

… ►The associated Coulomb–Laguerre polynomials are defined as … ►For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge. … ►

§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods

… ►

The Coulomb–Pollaczek Polynomials

… ►Full expressions for both

A_{x_{i},l}

and

B_{l}(x)

are given in Yamani and Reinhardt (1975) and it is seen that

|\ifrac{A_{x_{i},l}}{B_{l}(x_{i})}|^{2}

\ifrac{w_{i}^{N}}{w^{\mathrm{CP}}(x_{i})}

where

w^{N}_{i}

is the Gaussian-Pollaczek quadrature weight at

x=x_{i}

, and

w^{\mathrm{CP}}(x_{i})

is the Gaussian-Pollaczek weight function at the same quadrature abscissa. …