Gauss–Chebyshev formula

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1: 3.5 Quadrature

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Gauss–Chebyshev Formula

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2: 18.5 Explicit Representations

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Chebyshev

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§18.5(ii) Rodrigues Formulas

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

Chebyshev

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3: 18.12 Generating Functions

… ►and a similar formula by symmetry; compare the second row in Table 18.6.1. For the hypergeometric function ${}_{2}F_1$ see §§15.1, 15.2(i). … ►

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

Encodings:

pMML, png, TeX

See also:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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4: 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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5: 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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6: 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 $P_{\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{2}\mathrm{+}i\mathit{}\tau }^m\mathit{}\mathrm{(}z\mathrm{)}$ , Vyčisl. Centr Akad Nauk SSSR, Moscow. 1962

External Links:

MathReview, ZentralBlatt

Referenced by:

§14.20(vii).

Encodings:

BibTeX

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7: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Chebyshev Polynomials. Chapman & Hall/CRC, Boca Raton, FL.

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External Links:

ISBN 0-8493-0355-9, MathReview, ZentralBlatt

Referenced by:

§18.1(iii), §18.3, §18.3, Table 18.3.1, §18.38(i), §18.38(i), §18.40, (18.5.3), (18.5.4), Table 18.5.1, Table 18.6.1, (18.7.17), (18.7.18), (18.7.5), (18.7.6), (18.9.11), (18.9.12), §3.11(ii), §3.11(ii), §3.11(ii), §3.3(vi), §3.5(iv), Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

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Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. In Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), Vol. 49, pp. 169–178.

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External Links:

ISSN 0377-0427, Document, Link, MathReview Entry

Referenced by:

Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

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On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.

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External Links:

ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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The $q$-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.

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External Links:

ISSN 0035-7596, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§5.18(ii).

Encodings:

BibTeX

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