Chebyshev polynomials

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31: Bibliography B

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

Referenced by:

§8.7.

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32: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta \left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

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33: Bibliography D

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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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34: 8.27 Approximations

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Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\mathrm{\Gamma }(a,\omega z)$ (by specifying parameters) with , and $\gamma (a,\omega z)$ with $0\le \omega \le 1$; see also Temme (1994b, §3).

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Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the $z$-plane that exclude $z=0$ and are valid for .

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Luke (1975, p. 103) gives Chebyshev-series expansions for $E_1\left(x\right)$ and related functions for $x\ge 5$.

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Verbeeck (1970) gives polynomial and rational approximations for $E_p\left(x\right)=({\mathrm{e}}^{-x}/x)P(z)$, approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$.

35: Bibliography T

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Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.

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External Links:

ISBN 0-940600-05-6, FullText, MathReview (A. W. Davis), ZentralBlatt

Referenced by:

§35.4(i), §35.9.

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Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

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Referenced by:

§2.4(vi).

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Asymptotic estimates for Laguerre polynomials. Z. Angew. Math. Phys. 41 (1), pp. 114–126.

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ISSN 0044-2275, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§18.11(i), §18.16(vi).

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Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.

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ISSN 0922-5366, FullText, MathReview (Michael Hauss), ZentralBlatt

Referenced by:

§24.11, §24.16(i).

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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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Document

Referenced by:

§25.21(vii).

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36: 11.15 Approximations

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§11.15(i) Expansions in Chebyshev Series

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Luke (1975, pp. 416–421) gives Chebyshev-series expansions for ${𝐇}_n\left(x\right)$, ${𝐋}_n\left(x\right)$, $0\le \left|x\right|\le 8$, and ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, $x\ge 8$, for $n=0,1$; ${\int }_0^x{t}^{-m}{𝐇}_0\left(t\right)dt$, ${\int }_0^x{t}^{-m}{𝐋}_0\left(t\right)dt$, $0\le \left|x\right|\le 8$, $m=0,1$ and ${\int }_0^x({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, $x\ge 8$; the coefficients are to 20D.

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MacLeod (1993) gives Chebyshev-series expansions for ${𝐋}_0\left(x\right)$, ${𝐋}_1\left(x\right)$, $0\le x\le 16$, and $I_0\left(x\right)-{𝐋}_0\left(x\right)$, $I_1\left(x\right)-{𝐋}_1\left(x\right)$, $x\ge 16$; the coefficients are to 20D.

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§11.15(ii) Rational and Polynomial Approximations

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Newman (1984) gives polynomial approximations for ${𝐇}_n\left(x\right)$ for $n=0,1$, $0\le x\le 3$, and rational-fraction approximations for ${𝐇}_n\left(x\right)-Y_n\left(x\right)$ for $n=0,1$, $x\ge 3$. The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.

37: 19.38 Approximations

… ►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. …Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. …

38: 5.23 Approximations

… ►Hart et al. (1968) gives minimax polynomial and rational approximations to $\mathrm{\Gamma }\left(x\right)$ and $\ln \mathrm{\Gamma }\left(x\right)$ in the intervals $0\le x\le 1$, $8\le x\le 1000$, $12\le x\le 1000$; precision is variable. … ►

§5.23(ii) Expansions in Chebyshev Series

►Luke (1969b) gives the coefficients to 20D for the Chebyshev-series expansions of $\mathrm{\Gamma }\left(1+x\right)$, $1/\mathrm{\Gamma }\left(1+x\right)$, $\mathrm{\Gamma }\left(x+3\right)$, $\ln \mathrm{\Gamma }\left(x+3\right)$, $\psi \left(x+3\right)$, and the first six derivatives of $\psi \left(x+3\right)$ for $0\le x\le 1$. …Clenshaw (1962) also gives 20D Chebyshev-series coefficients for $\mathrm{\Gamma }\left(1+x\right)$ and its reciprocal for $0\le x\le 1$. …

39: Bibliography W

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Global asymptotics of the Meixner polynomials. Asymptotic Analysis 75 (3-4), pp. 211–231.

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External Links:

ISSN 0921-7134, Document, MathReview (Mario Pérez Riera), ZentralBlatt

Referenced by:

§18.24, §18.24.

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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Rational Chebyshev approximation. Numer. Math. 10 (4), pp. 289–306.

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External Links:

ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.11(iii).

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Rational Chebyshev approximations for the Bessel functions $J_0(x)$, $J_1(x)$, $Y_0(x)$, $Y_1(x)$ . Math. Comp. 39 (160), pp. 617–623.

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Notes:

Tables on microfiche

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

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9th item.

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

Encodings:

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40: 7.24 Approximations

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§7.24(ii) Expansions in Chebyshev Series

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Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi }\mathrm{erf}x$ and ${\mathrm{e}}^{x^2}F\left(x\right)$ for $-3\le x\le 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi }x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ and $2xF\left(x\right)$ for $x\ge 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

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Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ for $x\ge 3$ (15D).

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Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{-1}\mathrm{erf}x$ on $0\le x\le 2$, for $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[2,\mathrm{\infty })$, and for ${\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[0,\mathrm{\infty })$ (30D).

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Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x){\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $(0,\mathrm{\infty })$ (22D).

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