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Chebyshev polynomials

Table of contents: Definitions - Tables - Specific values - Zeros and extrema - Symmetries - Orthogonality - Differential equations - Recurrence relations - Order transformations - Trigonometric formulas - Power formulas - Product representations - Hypergeometric representations - Generating functions - Bounds and inequalities

Definitions

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Symbol: ChebyshevT Tn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
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Symbol: ChebyshevU Un ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind

Tables

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Table of Tn ⁣(x)T_{n}\!\left(x\right) for 0n150 \le n \le 15
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Table of Un ⁣(x)U_{n}\!\left(x\right) for 0n150 \le n \le 15

Specific values

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T0 ⁣(x)=1T_{0}\!\left(x\right) = 1
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T1 ⁣(x)=xT_{1}\!\left(x\right) = x
48765b
U0 ⁣(x)=1U_{0}\!\left(x\right) = 1
75eacb
U1 ⁣(x)=2xU_{1}\!\left(x\right) = 2 x
9001e6
U1 ⁣(x)=0U_{-1}\!\left(x\right) = 0
fc5d42
Tn ⁣(1)=1T_{n}\!\left(1\right) = 1
2760e7
Tn ⁣(1)=(1)nT_{n}\!\left(-1\right) = {\left(-1\right)}^{n}
a46d91
T2n ⁣(0)=(1)nT_{2 n}\!\left(0\right) = {\left(-1\right)}^{n}
42102c
T2n+1 ⁣(0)=0T_{2 n + 1}\!\left(0\right) = 0
e03fa4
Un ⁣(1)=n+1U_{n}\!\left(1\right) = n + 1
be9a45
Un ⁣(1)=(1)n(n+1)U_{n}\!\left(-1\right) = {\left(-1\right)}^{n} \left(n + 1\right)
2a5337
U2n ⁣(0)=(1)nU_{2 n}\!\left(0\right) = {\left(-1\right)}^{n}
7d111e
U2n+1 ⁣(0)=0U_{2 n + 1}\!\left(0\right) = 0

Zeros and extrema

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zerosxCTn ⁣(x)={cos ⁣(2k12nπ):k{1,2,n}}\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} T_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{2 k - 1}{2 n} \pi\right) : k \in \{1, 2, \ldots n\} \right\}
ce39ac
zerosxCUn ⁣(x)={cos ⁣(kn+1π):k{1,2,n}}\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} U_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{k}{n + 1} \pi\right) : k \in \{1, 2, \ldots n\} \right\}
3d25dd
solutionsxC[Tn ⁣(x){1,1}]={cos ⁣(knπ):k{0,1,n}}\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) \in \left\{-1, 1\right\}\right] = \left\{ \cos\!\left(\frac{k}{n} \pi\right) : k \in \{0, 1, \ldots n\} \right\}
b5a25e
solutionsxC[Tn ⁣(x)=1]={cos ⁣(2knπ):k{0,1,n2}}\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) = 1\right] = \left\{ \cos\!\left(\frac{2 k}{n} \pi\right) : k \in \{0, 1, \ldots \left\lfloor \frac{n}{2} \right\rfloor\} \right\}
db2b0a
solutionsxC[Tn ⁣(x)=1]={cos ⁣(2k1nπ):k{1,2,n+12}}\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) = -1\right] = \left\{ \cos\!\left(\frac{2 k - 1}{n} \pi\right) : k \in \{1, 2, \ldots \left\lfloor \frac{n + 1}{2} \right\rfloor\} \right\}

Symmetries

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Tn ⁣(x)=(1)nTn ⁣(x)T_{n}\!\left(-x\right) = {\left(-1\right)}^{n} T_{n}\!\left(x\right)
88aeb6
Un ⁣(x)=(1)nUn ⁣(x)U_{n}\!\left(-x\right) = {\left(-1\right)}^{n} U_{n}\!\left(x\right)
9093a3
Tn ⁣(x)=Tn ⁣(x)T_{-n}\!\left(x\right) = T_{n}\!\left(x\right)
78f5bb
Un ⁣(x)=Un2 ⁣(x)U_{-n}\!\left(x\right) = -U_{n - 2}\!\left(x\right)

Orthogonality

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11Tn ⁣(x)Tm ⁣(x)11x2dx=π2(δ(n,m)+δ(n,0)δ(m,0))\int_{-1}^{1} T_{n}\!\left(x\right) T_{m}\!\left(x\right) \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = \frac{\pi}{2} \left(\delta_{(n,m)} + \delta_{(n,0)} \delta_{(m,0)}\right)
473c36
11Un ⁣(x)Um ⁣(x)1x2dx=π2δ(n,m)\int_{-1}^{1} U_{n}\!\left(x\right) U_{m}\!\left(x\right) \sqrt{1 - {x}^{2}} \, dx = \frac{\pi}{2} \delta_{(n,m)}

Differential equations

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(1x2)y(x)xy(x)+n2y ⁣(x)=0   where y ⁣(x)=c1Tn ⁣(x)+c2Un1 ⁣(x)1x2\left(1 - {x}^{2}\right) y''(x) - x y'(x) + {n}^{2} y\!\left(x\right) = 0\; \text{ where } y\!\left(x\right) = {c}_{1} T_{n}\!\left(x\right) + {c}_{2} U_{n - 1}\!\left(x\right) \sqrt{1 - {x}^{2}}
30b67b
(1x2)y(x)3xy(x)+n(n+2)y ⁣(x)=0   where y ⁣(x)=c1Un ⁣(x)+c2Tn+1 ⁣(x)1x2\left(1 - {x}^{2}\right) y''(x) - 3 x y'(x) + n \left(n + 2\right) y\!\left(x\right) = 0\; \text{ where } y\!\left(x\right) = {c}_{1} U_{n}\!\left(x\right) + {c}_{2} \frac{T_{n + 1}\!\left(x\right)}{\sqrt{1 - {x}^{2}}}

Recurrence relations

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Tn ⁣(x)=2xTn1 ⁣(x)Tn2 ⁣(x)T_{n}\!\left(x\right) = 2 x T_{n - 1}\!\left(x\right) - T_{n - 2}\!\left(x\right)
d1ef91
Un ⁣(x)=2xUn1 ⁣(x)Un2 ⁣(x)U_{n}\!\left(x\right) = 2 x U_{n - 1}\!\left(x\right) - U_{n - 2}\!\left(x\right)
8a785a
Tn ⁣(x)=2xTn+1 ⁣(x)Tn+2 ⁣(x)T_{n}\!\left(x\right) = 2 x T_{n + 1}\!\left(x\right) - T_{n + 2}\!\left(x\right)
303204
Un ⁣(x)=2xUn+1 ⁣(x)Un+2 ⁣(x)U_{n}\!\left(x\right) = 2 x U_{n + 1}\!\left(x\right) - U_{n + 2}\!\left(x\right)
7b2c26
Tn ⁣(x)=xTn1 ⁣(x)(1x2)Un2 ⁣(x)T_{n}\!\left(x\right) = x T_{n - 1}\!\left(x\right) - \left(1 - {x}^{2}\right) U_{n - 2}\!\left(x\right)
ce5e03
Un ⁣(x)=xUn1 ⁣(x)+Tn ⁣(x)U_{n}\!\left(x\right) = x U_{n - 1}\!\left(x\right) + T_{n}\!\left(x\right)
0649c9
Tn ⁣(x)=Un ⁣(x)Un2 ⁣(x)2T_{n}\!\left(x\right) = \frac{U_{n}\!\left(x\right) - U_{n - 2}\!\left(x\right)}{2}
844561
Tn ⁣(x)=Un ⁣(x)xUn1 ⁣(x)T_{n}\!\left(x\right) = U_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)

Order transformations

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Tm ⁣(Tn ⁣(x))=Tmn ⁣(x)T_{m}\!\left(T_{n}\!\left(x\right)\right) = T_{m n}\!\left(x\right)
ed5222
Tm ⁣(x)Tn ⁣(x)=Tm+n ⁣(x)+Tmn ⁣(x)2T_{m}\!\left(x\right) T_{n}\!\left(x\right) = \frac{T_{m + n}\!\left(x\right) + T_{\left|m - n\right|}\!\left(x\right)}{2}
4b83c6
T2n ⁣(x)=2(Tn ⁣(x))21T_{2 n}\!\left(x\right) = 2 {\left(T_{n}\!\left(x\right)\right)}^{2} - 1
de0968
T2n+1 ⁣(x)=2Tn+1 ⁣(x)Tn ⁣(x)xT_{2 n + 1}\!\left(x\right) = 2 T_{n + 1}\!\left(x\right) T_{n}\!\left(x\right) - x
82288c
T2n ⁣(x)=Tn ⁣(2x21)T_{2 n}\!\left(x\right) = T_{n}\!\left(2 {x}^{2} - 1\right)
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U2n ⁣(x)=Tn ⁣(2x21)+Un1 ⁣(2x21)U_{2 n}\!\left(x\right) = T_{n}\!\left(2 {x}^{2} - 1\right) + U_{n - 1}\!\left(2 {x}^{2} - 1\right)

Trigonometric formulas

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Tn ⁣(x)=cos ⁣(nacos ⁣(x))T_{n}\!\left(x\right) = \cos\!\left(n \operatorname{acos}\!\left(x\right)\right)
2fc479
Tn ⁣(x)=cosh ⁣(nacosh ⁣(x))T_{n}\!\left(x\right) = \cosh\!\left(n \operatorname{acosh}\!\left(x\right)\right)
b8fdcd
Un1 ⁣(x)1x2=sin ⁣(nacos ⁣(x))U_{n - 1}\!\left(x\right) \sqrt{1 - {x}^{2}} = \sin\!\left(n \operatorname{acos}\!\left(x\right)\right)
f4b3fa
Tn ⁣(cos ⁣(x))=cos ⁣(nx)T_{n}\!\left(\cos\!\left(x\right)\right) = \cos\!\left(n x\right)
4c7aeb
Un ⁣(cos ⁣(x))sin ⁣(x)=sin ⁣(nx)U_{n}\!\left(\cos\!\left(x\right)\right) \sin\!\left(x\right) = \sin\!\left(n x\right)

Power formulas

0cbe75
Tn ⁣(x)=12((x+x21)n+(xx21)n)T_{n}\!\left(x\right) = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} + {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)
61375f
Un1 ⁣(x)x21=12((x+x21)n(xx21)n)U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} - {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)
fdf80d
Tn ⁣(x)+Un1 ⁣(x)x21=(x+x21)nT_{n}\!\left(x\right) + U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = {\left(x + \sqrt{{x}^{2} - 1}\right)}^{n}
42eb01
(Tn ⁣(x))2+(x21)(Un1 ⁣(x))2=1{\left(T_{n}\!\left(x\right)\right)}^{2} + \left({x}^{2} - 1\right) {\left(U_{n - 1}\!\left(x\right)\right)}^{2} = 1
5bd0ec
Tn ⁣(x+x12)=xn+xn2T_{n}\!\left(\frac{x + {x}^{-1}}{2}\right) = \frac{{x}^{n} + {x}^{-n}}{2}

Product representations

305a29
Tn ⁣(x)=2n1k=1n(xcos ⁣(2k12nπ))T_{n}\!\left(x\right) = {2}^{n - 1} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{2 k - 1}{2 n} \pi\right)\right)
f5fa23
Un ⁣(x)=2nk=1n(xcos ⁣(kn+1π))U_{n}\!\left(x\right) = {2}^{n} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{k}{n + 1} \pi\right)\right)

Hypergeometric representations

382679
Tn ⁣(x)=2F1 ⁣(n,n,12,1x2)T_{n}\!\left(x\right) = \,{}_2F_1\!\left(-n, n, \frac{1}{2}, \frac{1 - x}{2}\right)
ce9a39
Un ⁣(x)=(n+1)2F1 ⁣(n,n+2,32,1x2)U_{n}\!\left(x\right) = \left(n + 1\right) \,{}_2F_1\!\left(-n, n + 2, \frac{3}{2}, \frac{1 - x}{2}\right)

Generating functions

685d1a
n=0Tn ⁣(x)zn=1xz12xz+z2\sum_{n=0}^{\infty} T_{n}\!\left(x\right) {z}^{n} = \frac{1 - x z}{1 - 2 x z + {z}^{2}}
b5049d
n=0Un ⁣(x)zn=112xz+z2\sum_{n=0}^{\infty} U_{n}\!\left(x\right) {z}^{n} = \frac{1}{1 - 2 x z + {z}^{2}}
27b2bb
n=1Tn ⁣(x)znn=12log ⁣(12xz+z2)\sum_{n=1}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n} = -\frac{1}{2} \log\!\left(1 - 2 x z + {z}^{2}\right)
9d7c61
n=0Tn ⁣(x)znn!=ezxcosh ⁣(zx21)\sum_{n=0}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \cosh\!\left(z \sqrt{{x}^{2} - 1}\right)
fff8ff
n=0Un ⁣(x)znn!=ezx(cosh ⁣(zx21)+zxsinc ⁣(izx21))\sum_{n=0}^{\infty} U_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \left(\cosh\!\left(z \sqrt{{x}^{2} - 1}\right) + z x \operatorname{sinc}\!\left(i z \sqrt{{x}^{2} - 1}\right)\right)

Bounds and inequalities

15dd69
Tn ⁣(x)1\left|T_{n}\!\left(x\right)\right| \le 1
3c662e
Un ⁣(x)n+1\left|U_{n}\!\left(x\right)\right| \le \left|n + 1\right|
c718ea
Tn ⁣(z)Tn ⁣(iz)\left|T_{n}\!\left(z\right)\right| \le \left|T_{n}\!\left(i \left|z\right|\right)\right|
0b3fd6
Un ⁣(z)Un ⁣(iz)\left|U_{n}\!\left(z\right)\right| \le \left|U_{n}\!\left(i \left|z\right|\right)\right|
443759
Tn ⁣(z)(z+z2+1)n\left|T_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}
2a4b9d
Un ⁣(z)(z+z2+1)n\left|U_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-07-15 23:42:41.550119 UTC