Chebyshev

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11: 18.9 Recurrence Relations and Derivatives

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Chebyshev

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18.9.9 $$T_n\left(x\right)=\frac{1}{2}\left(U_n\left(x\right)-U_{n-2}\left(x\right)\right),$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.8

Referenced by:

§18.7(i), §18.9(ii)

Permalink:

http://dlmf.nist.gov/18.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

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18.9.12 $$T_{n+1}\left(x\right)+T_n\left(x\right)=(1+x)V_n\left(x\right).$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.7(i), §18.9(ii), §18.9(ii), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.9.E12

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, on the right-hand side we replaced $(1+x)W_n\left(x\right)$ with $(1+x)V_n\left(x\right)$. For further details see Errata.

See also:

Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

►Identities similar to (18.9.11) and (18.9.12) involving $W_n\left(x\right)$ and $T_n\left(x\right)$ can be obtained using rows 4 and 7 in Table 18.6.1. … ►

Chebyshev

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12: 9.19 Approximations

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§9.19(ii) Expansions in Chebyshev Series

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Prince (1975) covers $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$. The Chebyshev coefficients are given to 10-11D. Fortran programs are included. See also Razaz and Schonfelder (1981).

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Németh (1992, Chapter 8) covers $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, and integrals ${\int }_0^x\mathrm{Ai}\left(t\right)dt$, ${\int }_0^x\mathrm{Bi}\left(t\right)dt$, ${\int }_0^x{\int }_0^v\mathrm{Ai}\left(t\right)dtdv$, ${\int }_0^x{\int }_0^v\mathrm{Bi}\left(t\right)dtdv$ (see also (9.10.20) and (9.10.21)). The Chebyshev coefficients are given to 15D. Chebyshev coefficients are also given for expansions of the second and higher (real) zeros of $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, again to 15D.

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Razaz and Schonfelder (1980) covers $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$. The Chebyshev coefficients are given to 30D.

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MacLeod (1994) supplies Chebyshev-series expansions to cover $\mathrm{Gi}\left(x\right)$ for and $\mathrm{Hi}\left(x\right)$ for . The Chebyshev coefficients are given to 20D.

13: 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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14: 5.23 Approximations

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§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 $0\le x\le 1$. …

15: 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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16: 3.11 Approximation Techniques

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§3.11(ii) Chebyshev-Series Expansions

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Chebyshev Expansions

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Calculation of Chebyshev Coefficients

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Summation of Chebyshev Series: Clenshaw’s Algorithm

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17: 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:

pMML, png, TeX

See also:

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

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25.16.2 $$\psi \left(x\right)=x-\frac{{\zeta }^{\prime }\left(0\right)}{\zeta \left(0\right)}-\sum _{\rho }\frac{x^{\rho }}{\rho }+o\left(1\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$\psi \left(x\right)$: Chebyshev $\psi$-function, $\zeta \left(s\right)$: Riemann zeta function, $o\left(x\right)$: order less than and $x$: real variable

Keywords:

asymptotic approximation

Source:

Apostol (2000, p. 9)

Permalink:

http://dlmf.nist.gov/25.16.E2

Encodings:

pMML, png, TeX

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

Encodings:

pMML, png, TeX

See also:

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

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25.16.4 $$\psi \left(x\right)=x+O\left(x^{\frac{1}{2}+ϵ}\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\psi \left(x\right)$: Chebyshev $\psi$-function and $x$: real variable

Keywords:

asymptotic approximation

Source:

Apostol (2000, p. 9)

Permalink:

http://dlmf.nist.gov/25.16.E4

Encodings:

pMML, png, TeX

See also:

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

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18: 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 (1975, p. 103) gives Chebyshev-series expansions for $E_1\left(x\right)$ and related functions for $x\ge 5$.

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19: 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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20: 29.15 Fourier Series and Chebyshev Series

§29.15 Fourier Series and Chebyshev Series

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§29.15(ii) Chebyshev Series

►The Chebyshev polynomial $T$ of the first kind (§18.3) satisfies $\cos \left(p\varphi \right)=T_p\left(\cos \varphi \right)$. … ►

29.15.43 $${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)=\frac{1}{2}{A}_0+\sum _{p=1}^n{A}_{2p}{T}_{2p}\left(\mathrm{sn}(z,k)\right).$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $A_{2p}$: coefficients

Permalink:

http://dlmf.nist.gov/29.15.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

… ►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 …