Gauss–Chebyshev formula

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1: 3.5 Quadrature

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Gauss–Chebyshev Formula

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2: 18.3 Definitions

… ►Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). …

3: 18.5 Explicit Representations

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§18.5(ii) Rodrigues Formulas

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Chebyshev

… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite ${\mathrm{𝐻𝑒}}_n$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►

Chebyshev

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4: 18.38 Mathematical Applications

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§18.38(i) Classical OP’s: Numerical Analysis

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Approximation Theory

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Differential Equations: Spectral Methods

►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). … ►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas. …

5: 18.12 Generating Functions

… ►The $z$-radii of convergence will depend on $x$, and in first instance we will assume $x\in [-1,1]$ for Jacobi, ultraspherical, Chebyshev and Legendre, $x\in [0,\mathrm{\infty })$ for Laguerre, and $x\in ℝ$ for Hermite. … ►and similar formulas as (18.12.3) and (18.12.3_5) by symmetry; compare the second row in Table 18.6.1. … ►

Chebyshev

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18.12.7 $$\frac{1-z^2}{1-2xz+z^2}=1+2\sum _{n=1}^{\mathrm{\infty }}{T}_n\left(x\right)z^n,$$ .

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E7

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Annotations for §18.12, §18.12 and Ch.18

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18.12.8 $$\frac{1-xz}{1-2xz+z^2}=\sum _{n=0}^{\mathrm{\infty }}{T}_n\left(x\right)z^n,$$ .

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.6

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E8

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Annotations for §18.12, §18.12 and Ch.18

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6: Bibliography T

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On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.

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Notes:

Painlevé functions and asymptotic analysis (Japanese) (Kyoto, 1995)

External Links:

FullText, MathReview (Andrei A. Kapaev)

Referenced by:

§32.12(i).

Encodings:

BibTeX

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Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.

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External Links:

ISSN 0025-5858, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.4(v), §24.6.

Encodings:

BibTeX

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Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.

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External Links:

ISSN 0036-1445, Document, MathReview (Kai Diethelm), ZentralBlatt

Referenced by:

§3.5(iv).

Encodings:

BibTeX

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Rational Chebyshev approximation for the Fermi-Dirac integral $F_{-3/2}(x)$ . Solid–State Electronics 41 (5), pp. 771–773.

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External Links:

Document

Referenced by:

§25.21(vii).

Encodings:

BibTeX

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7: Bibliography R

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Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.

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External Links:

ISSN 0029-599X, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§3.11(i).

Encodings:

BibTeX

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High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

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Referenced by:

3rd item.

Encodings:

BibTeX

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Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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External Links:

ISSN 0098-3500, Document

Referenced by:

1st item.

Encodings:

BibTeX

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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

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Notes:

English translation, AEC-TR-4491, United States Atomic Energy Commission

External Links:

MathReview (A. S. Householder)

Referenced by:

§3.11(iii).

Encodings:

BibTeX

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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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8: Bibliography C

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Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée. Mem. Ac. Sc. St. Pétersbourg 6, pp. 141–157.

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Referenced by:

§25.16(i).

Encodings:

BibTeX

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A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9 (51), pp. 118–120.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§18.40(i).

Encodings:

BibTeX

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Chebyshev polynomial expansions of complete elliptic integrals. Math. Comp. 19 (90), pp. 249–259.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

BibTeX

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Chebyshev approximations for the Fresnel integrals. Math. Comp. 22 (102), pp. 450–453.

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Notes:

with Microfiche Supplement

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Rational Chebyshev approximations for the error function. Math. Comp. 23 (107), pp. 631–637.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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9: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

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Confluent Form

… ►For usage of the zeros of an OP in Gauss quadrature see §3.5(v). … ►This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with $p_n(x)=T_n\left(x\right)$, a Chebyshev polynomial of the first kind, see Table 18.3.1. … ►

Degree lowering and raising differentiation formulas and structure relations

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10: Bibliography Z

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Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.

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External Links:

ISSN 1021-2655, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§13.9(i), §15.13.

Encodings:

BibTeX

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Chebyshev series approximations for the Bessel function $Y_n(z)$ of complex argument. Appl. Math. Comput. 88 (2-3), pp. 275–286.

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External Links:

ISSN 0096-3003, Document, MathReview (Sven Ehrich), ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

BibTeX

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Tables and formulae for the spherical functions $P_{-1/2+i\tau }^m(z)$ . Translated by E. L. Albasiny, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy i formuly dlya sfericheskikh funktsii $P_{-1/2+i\tau }^m(z)$ , Vyčisl. Centr Akad Nauk SSSR, Moscow. 1962

External Links:

MathReview, ZentralBlatt

Referenced by:

§14.20(vii).

Encodings:

BibTeX

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