in series of Chebyshev polynomials

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21: Bibliography S

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

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Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity. 4th edition, Springer-Verlag, Berlin.

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Notes:

Volume 7 in Springer Series in Information Sciences.

External Links:

ZentralBlatt

Referenced by:

§27.16, §27.17.

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:

ZentralBlatt

Referenced by:

§12.13(i).

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Chebyshev approximation of $(1+2x)\exp (x^2)\mathrm{erfc}x$ in . Math. Comp. 36 (153), pp. 249–253.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

4th item.

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An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

Encodings:

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22: 18.5 Explicit Representations

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Chebyshev

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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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23: Bibliography D

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Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical Sciences, New York.

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External Links:

ISBN 0-9658703-2-4; 0-8218-2695-6, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§18.38(ii), §18.38(ii).

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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.

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ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt

Referenced by:

§18.32.

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Chebyshev expansion of the associated Legendre polynomial $P_L^M(x)$ . Comput. Phys. Comm. 18 (1), pp. 63–71.

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Notes:

Double-precision Fortran, maximum accuracy 25D

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Document, Code

Referenced by:

3rd item.

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Chebyshev series for the spherical Bessel function $j_l(r)$ . Comput. Phys. Comm. 18 (1), pp. 73–86.

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External Links:

Document

Referenced by:

§10.76(iii).

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Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§27.11.

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24: Bibliography M

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Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

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External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

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Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0770-6, MathReview (Angèle M. Hamel), ZentralBlatt

Referenced by:

§17.14.

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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(i), (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.

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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, ZentralBlatt

Referenced by:

Erratum (V1.0.28) for Table 18.3.1.

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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).

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25: Bibliography F

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Multivariate Calculation. Use of the Continuous Groups. Springer Series in Statistics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96049-X, MathReview (Yasuko Chikuse), ZentralBlatt

Referenced by:

§35.9.

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Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

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External Links:

ISSN 0022-2488, Document, MathReview (Shao-Ming Fei), ZentralBlatt

Referenced by:

§13.3(ii), §15.5(i), §16.3(i).

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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

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Notes:

Reprint with corrections of two books published originally in 1916 and 1936.

External Links:

MathReview

Referenced by:

§2.10(iii).

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Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London.

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Notes:

Reprinted in 1972 with corrections.

External Links:

MathReview (G. N. Lance), ZentralBlatt

Referenced by:

§3.11(ii).

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On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.32, §32.15.

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26: Bibliography C

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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).

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The numerical solution of linear differential equations in Chebyshev series. Proc. Cambridge Philos. Soc. 53 (1), pp. 134–149.

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External Links:

MathReview (L. Fox), ZentralBlatt

Referenced by:

§18.38(i), §3.11(ii).

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Chebyshev Series for Mathematical Functions. National Physical Laboratory Mathematical Tables, Vol. 5. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§3.11(ii), §4.47(i), §5.23(ii), 1st item.

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

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

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Chebyshev expansions for the Bessel function $J_n(z)$ in the complex plane. Math. Comp. 40 (161), pp. 343–366.

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External Links:

ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

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27: 25.16 Mathematical Applications

… ►In studying the distribution of primes $p\le x$, Chebyshev (1851) introduced a function $\psi \left(x\right)$ (not to be confused with the digamma function used elsewhere in this chapter), given by ►

25.16.1 $$\psi \left(x\right)=\sum _{m=1}^{\mathrm{\infty }}\sum _{p^m\le x}\ln p,$$

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Defines:

$\psi \left(x\right)$: Chebyshev $\psi$-function

Symbols:

$\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $p$: prime number and $x$: real variable

Keywords:

definition, double series, infinite series, series representation

Source:

Apostol (1976, p. 75)

Permalink:

http://dlmf.nist.gov/25.16.E1

Encodings:

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See also:

Annotations for §25.16(i), §25.16 and Ch.25

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25.16.3 $$\psi \left(x\right)=x+o\left(x\right),$$ $x\to \mathrm{\infty }$.

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Symbols:

$\psi \left(x\right)$: Chebyshev $\psi$-function, $\sim$: asymptotic equality, $o\left(x\right)$: order less than and $x$: real variable

Keywords:

asymptotic approximation

Source:

Apostol (1976, (10), p. 79); since $\psi \left(x\right)\sim x$ is equivalent to $\psi \left(x\right)=x+o\left(x\right)$

Referenced by:

§25.10(i)

Permalink:

http://dlmf.nist.gov/25.16.E3

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See also:

Annotations for §25.16(i), §25.16 and Ch.25

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25.16.6 $$H\left(s\right)=-{\zeta }^{\prime }\left(s\right)+\gamma \zeta \left(s\right)+\frac{1}{2}\zeta \left(s+1\right)+\sum _{r=1}^k\zeta \left(1-2r\right)\zeta \left(s+2r\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{2k+2}}dx,$$

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Symbols:

$\gamma$: Euler’s constant, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (3), p. 87)

Permalink:

http://dlmf.nist.gov/25.16.E6

Encodings:

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See also:

Annotations for §25.16(ii), §25.16 and Ch.25

… ►For integer $s$ ($\ge 2$), $H\left(s\right)$ can be evaluated in terms of the zeta function: …

28: 3.5 Quadrature

… ►For the latter $a=-1$, $b=1$, and the nodes $x_k$ are the extrema of the Chebyshev polynomial $T_n\left(x\right)$ (§3.11(ii) and §18.3). … ►

Gauss–Chebyshev Formula

… ►In the case of Chebyshev weight functions $w(x)={(1-x)}^{\alpha }{(1+x)}^{\beta }$ on $[-1,1]$, with $\left|\alpha \right|=\left|\beta \right|=\frac{1}{2}$, the nodes $x_k$, weights $w_k$, and error constant ${\gamma }_n$, are as follows: … … ►Below we give for the classical orthogonal polynomials the recurrence coefficients ${\alpha }_n$ and ${\beta }_n$ in (3.5.30). …

29: Bibliography B

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Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 25–53.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§18.38(iii).

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Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

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External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§8.7.

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Singularities in Waves and Rays. In Les Houches Lecture Series Session XXXV, R. Balian, M. Kléman, and J.-P. Poirier (Eds.), Vol. 35, pp. 453–543.

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Referenced by:

§36.14(i).

Encodings:

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Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0324-7, MathReview, ZentralBlatt

Referenced by:

§26.19.

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Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.

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External Links:

ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt

Referenced by:

§3.11(iv).

Encodings:

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