Chebyshev

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1: 18.3 Definitions

§18.3 Definitions

… ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). … ►

Chebyshev

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

2: 4.47 Approximations

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§4.47(i) Chebyshev-Series Expansions

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3: 6.20 Approximations

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§6.20(ii) Expansions in Chebyshev Series

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Clenshaw (1962) gives Chebyshev coefficients for $-E_1\left(x\right)-\ln |x|$ for $-4\le x\le 4$ and ${\mathrm{e}}^x{E}_1\left(x\right)$ for $x\ge 4$ (20D).

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Luke and Wimp (1963) covers $\mathrm{Ei}\left(x\right)$ for $x\le -4$ (20D), and $\mathrm{Si}\left(x\right)$ and $\mathrm{Ci}\left(x\right)$ for $x\ge 4$ (20D).

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Luke (1969b, pp. 321–322) covers $\mathrm{Ein}\left(x\right)$ and $-\mathrm{Ein}\left(-x\right)$ for $0\le x\le 8$ (the Chebyshev coefficients are given to 20D); $E_1\left(x\right)$ for $x\ge 5$ (20D), and $\mathrm{Ei}\left(x\right)$ for $x\ge 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function (§13.2(i)) from which Chebyshev expansions near infinity for $E_1\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If the scheme can be used in backward direction.

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4: 12.20 Approximations

§12.20 Approximations

►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U(a,b,x)$ and $M(a,b,x)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\mathrm{\infty }$ and $x=0$, respectively. As special cases of these results a Chebyshev-series expansion for $U(a,x)$ valid when follows from (12.7.14), and Chebyshev-series expansions for $U(a,x)$ and $V(a,x)$ valid when $0\le x\le \lambda$ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …

5: 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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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta \left(s+1\right)$ and $\zeta \left(s+k\right)$, $k=2,3,4,5,8$, for $0\le s\le 1$ (23D).

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Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta \left(s\right)$ for $0\le s\le 1$ (15D), $\zeta \left(s+1\right)$ for $0\le s\le 1$ (20D), and $\ln \xi \left(\frac{1}{2}+\mathrm{i}x\right)$ (§25.4) for $-1\le x\le 1$ (20D). For errata see Piessens and Branders (1972).

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Antia (1993) gives minimax rational approximations for $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals and , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors ${10}^{-4},{10}^{-8},{10}^{-12}$.

6: 18.41 Tables

… ►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_n\left(x\right)$, $U_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_n\left(x\right)$ and $U_n\left(x\right)$, and $0.5,1,3,5,10$ for $L_n\left(x\right)$ and $H_n\left(x\right)$. …

7: 16.26 Approximations

… ►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

8: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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Chebyshev, Ultraspherical, and Jacobi

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18.7.7 $$T_n^{\ast }\left(x\right)=T_n\left(2x-1\right),$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $T_n^{\ast }\left(x\right)$: shifted Chebyshev polynomial of the first kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.14

Referenced by:

§18.7(i)

Permalink:

http://dlmf.nist.gov/18.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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Ultraspherical $\to$ Chebyshev

… ► See §18.11(ii) for limit formulas of Mehler–Heine type.

9: 18.1 Notation

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Classical OP’s

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Chebyshev of first, second, third, and fourth kinds: $T_n\left(x\right)$, $U_n\left(x\right)$, $V_n\left(x\right)$, $W_n\left(x\right)$.

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Shifted Chebyshev of first and second kinds: $T_n^{\ast }\left(x\right)$, $U_n^{\ast }\left(x\right)$.

… ►Nor do we consider the shifted Jacobi polynomials: …or the dilated Chebyshev polynomials of the first and second kinds: …

10: 18.6 Symmetry, Special Values, and Limits to Monomials

… ►For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. … ►

Table 18.6.1: Classical OP’s: symmetry and special values.

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$p_n(x)$	$p_n(-x)$	$p_n(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime }(0)$
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$T_n\left(x\right)$	${(-1)}^n{T}_n\left(x\right)$	$1$	${(-1)}^n$	${(-1)}^n(2n+1)$
$U_n\left(x\right)$	${(-1)}^n{U}_n\left(x\right)$	$n+1$	${(-1)}^n$	${(-1)}^n(2n+2)$
$V_n\left(x\right)$	${(-1)}^n{W}_n\left(x\right)$	$1$	${(-1)}^n$	${(-1)}^n(2n+2)$
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