DLMF: Untitled Document

§18.3 Definitions

… ►

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

► ►►►►

Name	$p_{n}(x)$	$(a,b)$	$w(x)$	$h_{n}$	$k_{n}$	$\ifrac{\tilde{k}_{n}}{k_{n}}$
…
Chebyshev of second kind	$U_{n}\left(x\right)$	$(-1,1)$	$(1-x^{2})^{\frac{1}{2}}$	$\tfrac{1}{2}\pi$	$2^{n}$	$0$
…
Shifted Chebyshev of second kind	$U^{*}_{n}\left(x\right)$	$(0,1)$	$(x-x^{2})^{\frac{1}{2}}$	$\tfrac{1}{8}\pi$	$2^{2n}$	$-\tfrac{1}{2}n$
…

► … ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials

T_{n}\left(x\right)

,

n=0,1,\dots,N

, are orthogonal on the discrete point set comprising the zeros

x_{N+1,n},n=1,2,\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). …

… ►For

P_{n}\left(x\right)

(

=\mathsf{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)

. …

… ►

Chebyshev, Ultraspherical, and Jacobi

… ►

18.7.7

T^{*}_{n}\left(x\right)=T_{n}\left(2x-1\right),

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $T^{*}_{\NVar{n}}\left(\NVar{x}\right)$ : 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:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $U^{*}_{\NVar{n}}\left(\NVar{x}\right)$ : 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:: $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : 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)=xV_{n}\left(2x^{2}-1\right).

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : 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 $xW_{n}\left(2x^{2}-1\right)$ , with $xV_{n}\left(2x^{2}-1\right)$ . For further details see Errata.
See also:: Annotations for §18.7(ii), §18.7 and Ch.18

…

… ►

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

► ►►►►

$p_{n}(x)$	$A_{n}$	$B_{n}$	$C_{n}$
…
$U_{n}\left(x\right)$	$2$	$0$	$1$
…
$U^{*}_{n}\left(x\right)$	$4$	$-2$	$1$
…

► … ►

18.9.9

T_{n}\left(x\right)=\tfrac{1}{2}\left(U_{n}\left(x\right)-U_{n-2}\left(x\right% )\right),

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : 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

… ►

18.9.12

T_{n+1}\left(x\right)+T_{n}\left(x\right)=(1+x)V_{n}\left(x\right).

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : 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 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

►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 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(\tfrac{1}{2}x\right),

►

S_{n}\left(x\right)=U_{n}\left(\tfrac{1}{2}x\right).

…

… ►

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

► ►►►►►►

$p_{n}(x)$	$p_{n}(-x)$	$p_{n}(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime}(0)$
…
$T_{n}\left(x\right)$	$(-1)^{n}T_{n}\left(x\right)$	$1$	$(-1)^{n}$	$(-1)^{n}(2n+1)$
$U_{n}\left(x\right)$	$(-1)^{n}U_{n}\left(x\right)$	$n+1$	$(-1)^{n}$	$(-1)^{n}(2n+2)$
$V_{n}\left(x\right)$	$(-1)^{n}W_{n}\left(x\right)$	$1$	$(-1)^{n}$	$(-1)^{n}(2n+2)$
$W_{n}\left(x\right)$	$(-1)^{n}V_{n}\left(x\right)$	$2n+1$	$(-1)^{n}$	$(-1)^{n}(2n+2)$
…

► …

… ►

T_{0}\left(x\right)=1,

►

T_{1}\left(x\right)=x,

►

T_{2}\left(x\right)=2x^{2}-1,

►

T_{3}\left(x\right)=4x^{3}-3x,

… ►

U_{0}\left(x\right)=1,

…

… ►

§29.15(ii) Chebyshev Series

►The Chebyshev polynomial

T

of the first kind (§18.3) satisfies

\cos\left(p\phi\right)=T_{p}\left(\cos\phi\right)

. … ►

29.15.43

\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p% }T_{2p}\left(\operatorname{sn}\left(z,k\right)\right).

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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)\phi\right)=(\sin\phi)U_{p}\left(\cos\phi\right)

, with

U

denoting the Chebyshev polynomial of the second kind (§18.3), we obtain ►

29.15.44

\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)=\sum_{p=0}^{n}A_{2p+1}T_{2p+1}\left% (\operatorname{sn}\left(z,k\right)\right),

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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

…

… ►The Chebyshev polynomials

T_{n}

are given by … ►

3.11.7

T_{n+1}\left(x\right)-2xT_{n}\left(x\right)+T_{n-1}\left(x\right)=0,

n=1,2,\dots

,

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind
Referenced by:: §3.11(ii)
Permalink:: http://dlmf.nist.gov/3.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §3.11(ii), §3.11 and Ch.3

►with initial values

T_{0}\left(x\right)=1

,

T_{1}\left(x\right)=x

. … ►For the expansion (3.11.11), numerical values of the Chebyshev polynomials

T_{n}\left(x\right)

can be generated by application of the recurrence relation (3.11.7). …Let

c_{n}T_{n}\left(x\right)

be the last term retained in the truncated series. …

… ►

Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

.

► ►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
5	$T_{n}\left(x\right)$	$1-x^{2}$	$-x$	$0$	$n^{2}$
6	$U_{n}\left(x\right)$	$1-x^{2}$	$-3x$	$0$	$n(n+2)$
…

► …

Chebyshev polynomials

1—10 of 47 matching pages

1: 18.3 Definitions

§18.3 Definitions

2: 18.41 Tables

3: 18.7 Interrelations and Limit Relations

Chebyshev, Ultraspherical, and Jacobi

4: 18.9 Recurrence Relations and Derivatives

5: 18.1 Notation

6: 18.6 Symmetry, Special Values, and Limits to Monomials

7: 18.5 Explicit Representations

8: 29.15 Fourier Series and Chebyshev Series

§29.15(ii) Chebyshev Series

9: 3.11 Approximation Techniques

10: 18.8 Differential Equations