Chebyshev polynomials

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1: 18.3 Definitions

§18.3 Definitions

… ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …

► ►►►►

Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
…
Chebyshev of second kind	$U_n\left(x\right)$	$(-1,1)$	${(1-x^2)}^{\frac{1}{2}}$	$\frac{1}{2}\pi$	$2^n$	$0$
…
Shifted Chebyshev of second kind	$U_n^{*}\left(x\right)$	$(0,1)$	${(x-x^2)}^{\frac{1}{2}}$	$\frac{1}{8}\pi$	$2^{2n}$	$-\frac{1}{2}n$
…

►

… ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_n\left(x\right)$, $n=0,1,\mathrm{\dots },N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\mathrm{\dots },N+1$, of $T_{N+1}\left(x\right)$: … ►For another version of the discrete orthogonality property of the polynomials $T_n\left(x\right)$ see (3.11.9). …

2: 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 $H_n\left(x\right)$. …

3: 18.9 Recurrence Relations and Derivatives

… ►

Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).

► ►►►►

$p_n(x)$	$A_n$	$B_n$	$C_n$
…
$U_n\left(x\right)$	$2$	$0$	$1$
…
$U_n^{*}\left(x\right)$	$4$	$-2$	$1$
…

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… ►

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

ⓘ

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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18.9.11 $$V_n\left(x\right)+V_{n-1}\left(x\right)=2T_n\left(x\right),$$

ⓘ

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:

Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.9.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshevpolynomials 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 left-hand side we replaced $W_n\left(x\right)+W_{n-1}\left(x\right)$ with $V_n\left(x\right)+V_{n-1}\left(x\right)$. For further details see Errata.

See also:

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

►

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

ⓘ

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), 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 Chebyshevpolynomials 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

…

4: 18.7 Interrelations and Limit Relations

… ►

Chebyshev, Ultraspherical, and Jacobi

… ►

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

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $T_n^{*}\left(x\right)$: 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

►

18.7.8 $$U_n^{*}\left(x\right)=U_n\left(2x-1\right).$$

ⓘ

Symbols:

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

A&S Ref:

22.5.15

Permalink:

http://dlmf.nist.gov/18.7.E8

Encodings:

pMML, png, TeX

See also:

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

… ►

18.7.17 $$U_{2n}\left(x\right)=W_n\left(2x^2-1\right),$$

ⓘ

Symbols:

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

A&S Ref:

22.5.30

Referenced by:

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

Permalink:

http://dlmf.nist.gov/18.7.E17

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshevpolynomials 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 $V_n\left(2x^2-1\right)$, with $W_n\left(2x^2-1\right)$. For further details see Errata.

See also:

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

►

18.7.18 $$T_{2n+1}\left(x\right)=x{V}_n\left(2x^2-1\right).$$

ⓘ

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

A&S Ref:

22.5.29

Referenced by:

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

Permalink:

http://dlmf.nist.gov/18.7.E18

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshevpolynomials 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 $x{W}_n\left(2x^2-1\right)$, with $x{V}_n\left(2x^2-1\right)$. For further details see Errata.

See also:

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

…

5: 18.5 Explicit Representations

… ►

$T_0\left(x\right)=1,$

►

$T_1\left(x\right)=x,$

►

$T_2\left(x\right)=2x^2-1,$

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$T_3\left(x\right)=4x^3-3x,$

… ►

$U_0\left(x\right)=1,$

…

6: 18.1 Notation

… ►

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

►

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

… ►Nor do we consider the shifted Jacobi polynomials: … ►

$C_n\left(x\right)=2T_n\left(\frac{1}{2}x\right),$

►

$S_n\left(x\right)=U_n\left(\frac{1}{2}x\right).$

…

7: 18.6 Symmetry, Special Values, and Limits to Monomials

… ►

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

► ►►►►►►

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

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…

8: 18.13 Continued Fractions

… ►

Chebyshev

► $T_n\left(x\right)$ is the denominator of the $n$th approximant to: …and $U_n\left(x\right)$ is the denominator of the $n$th approximant to: …

9: 29.15 Fourier Series and Chebyshev Series

… ►

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

ⓘ

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 ►

29.15.44 $${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)=\sum _{p=0}^n{A}_{2p+1}{T}_{2p+1}\left(\mathrm{sn}(z,k)\right),$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: 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.E44

Encodings:

pMML, png, TeX

See also:

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

…

10: 18.18 Sums

… ►

Chebyshev

►

18.18.21 $$T_m\left(x\right)T_n\left(x\right)=\frac{1}{2}(T_{m+n}\left(x\right)+T_{m-n}\left(x\right)).$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/18.18.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(v), §18.18(v), §18.18 and Ch.18

… ►

18.18.32 $$2\sum _{\mathrm{\ell }=0}^n{T}_{2\mathrm{\ell }}\left(x\right)=1+U_{2n}\left(x\right),$$

ⓘ

Symbols:

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

A&S Ref:

22.12.2

Referenced by:

§18.18(viii)

Permalink:

http://dlmf.nist.gov/18.18.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

►

18.18.33 $$2\sum _{\mathrm{\ell }=0}^n{T}_{2\mathrm{\ell }+1}\left(x\right)=U_{2n+1}\left(x\right).$$

ⓘ

Symbols:

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

A&S Ref:

22.12.3

Permalink:

http://dlmf.nist.gov/18.18.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

►

18.18.34 $$2(1-x^2)\sum _{\mathrm{\ell }=0}^n{U}_{2\mathrm{\ell }}\left(x\right)=1-T_{2n+2}\left(x\right),$$

ⓘ

Symbols:

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

A&S Ref:

22.12.4

Permalink:

http://dlmf.nist.gov/18.18.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

…