in series of Chebyshev polynomials

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1: 18.40 Methods of Computation

… ►For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. … …

2: 3.11 Approximation Techniques

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§3.11(ii) Chebyshev-Series Expansions

… ►Since $L_0=1$, $L_n$ is a monotonically increasing function of $n$, and (for example) $L_{1000}=4.07\mathrm{\dots }$, this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile. … ►Let $c_n{T}_n\left(x\right)$ be the last term retained in the truncated series. …

3: 18.18 Sums

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Chebyshev

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Chebyshev

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Legendre and Chebyshev

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4: 18.38 Mathematical Applications

… ►In consequence, expansions of functions that are infinitely differentiable on $[-1,1]$ in series of Chebyshev polynomials usually converge extremely rapidly. … ►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). …

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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6: 18.3 Definitions

… ►Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for $n=0,1,\mathrm{\dots },6$ are given in §18.5(iv). …

7: 29.15 Fourier Series and Chebyshev Series

§29.15 Fourier Series and Chebyshev Series

… ►When $\nu =2n$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.1) terminates: … ►

§29.15(ii) Chebyshev Series

►The Chebyshev polynomial $T$ of the first kind (§18.3) satisfies $\cos \left(p\varphi \right)=T_p\left(\cos \varphi \right)$. … ►Using also $\sin \left((p+1)\varphi \right)=(\sin \varphi )U_p\left(\cos \varphi \right)$, with $U$ denoting the Chebyshev polynomial of the second kind (§18.3), we obtain …

8: 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. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. ►Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

9: 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 $z=x^{-1}$.

10: 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 $0\le x\le 1$. These coefficients are reproduced in Luke (1975). Clenshaw (1962) also gives 20D Chebyshev-series coefficients for $\mathrm{\Gamma }\left(1+x\right)$ and its reciprocal for $0\le x\le 1$. …