Fungrim home page

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

1a0c43
Symbol: ChebyshevT — Tn ⁣(x)T_{n}\!\left(x\right) — Chebyshev polynomial of the first kind
d4e9aa
Symbol: ChebyshevU — Un ⁣(x)U_{n}\!\left(x\right) — Chebyshev polynomial of the second kind

Tables

85e42e
Table of Tn ⁣(x)T_{n}\!\left(x\right) for 0≤n≤150 \le n \le 15
fd8310
Table of Un ⁣(x)U_{n}\!\left(x\right) for 0≤n≤150 \le n \le 15

Specific values

c76e72
T0 ⁣(x)=1T_{0}\!\left(x\right) = 1
be5652
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
U−1 ⁣(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

7a7d1d
zeros⁡ x∈CTn ⁣(x)={cos⁡ ⁣(2k−12nπ):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
zeros⁡ x∈CUn ⁣(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
solutions⁡ x∈C[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
solutions⁡ x∈C[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
solutions⁡ x∈C[Tn ⁣(x)=−1]={cos⁡ ⁣(2k−1nπ):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

6a24ab
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
T−n ⁣(x)=Tn ⁣(x)T_{-n}\!\left(x\right) = T_{n}\!\left(x\right)
78f5bb
U−n ⁣(x)=−Un−2 ⁣(x)U_{-n}\!\left(x\right) = -U_{n - 2}\!\left(x\right)

Orthogonality

2c26a1
∫−11Tn ⁣(x)Tm ⁣(x)11−x2 dx=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)1−x2 dx=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

0ed026
(1−x2)y′′(x)−xy′(x)+n2y ⁣(x)=0   where y ⁣(x)=c1Tn ⁣(x)+c2Un−1 ⁣(x)1−x2\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
(1−x2)y′′(x)−3xy′(x)+n(n+2)y ⁣(x)=0   where y ⁣(x)=c1Un ⁣(x)+c2Tn+1 ⁣(x)1−x2\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

faeed9
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)
d1ef91
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)
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)=xTn−1 ⁣(x)−(1−x2)Un−2 ⁣(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)=xUn−1 ⁣(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)−Un−2 ⁣(x)2T_{n}\!\left(x\right) = \frac{U_{n}\!\left(x\right) - U_{n - 2}\!\left(x\right)}{2}
844561
Tn ⁣(x)=Un ⁣(x)−xUn−1 ⁣(x)T_{n}\!\left(x\right) = U_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)

Order transformations

7e882c
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)+T∣m−n∣ ⁣(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))2−1T_{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 ⁣(2x2−1)T_{2 n}\!\left(x\right) = T_{n}\!\left(2 {x}^{2} - 1\right)
5f09f4
U2n ⁣(x)=Tn ⁣(2x2−1)+Un−1 ⁣(2x2−1)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

fda800
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
Un−1 ⁣(x)1−x2=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+x2−1)n+(x−x2−1)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
Un−1 ⁣(x)x2−1=12((x+x2−1)n−(x−x2−1)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)+Un−1 ⁣(x)x2−1=(x+x2−1)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+(x2−1)(Un−1 ⁣(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+x−12)=xn+x−n2T_{n}\!\left(\frac{x + {x}^{-1}}{2}\right) = \frac{{x}^{n} + {x}^{-n}}{2}

Product representations

305a29
Tn ⁣(x)=2n−1∏k=1n(x−cos⁡ ⁣(2k−12nπ))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)=2n∏k=1n(x−cos⁡ ⁣(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,12,1−x2)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+2,32,1−x2)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=0∞Tn ⁣(x)zn=1−xz1−2xz+z2\sum_{n=0}^{\infty} T_{n}\!\left(x\right) {z}^{n} = \frac{1 - x z}{1 - 2 x z + {z}^{2}}
b5049d
∑n=0∞Un ⁣(x)zn=11−2xz+z2\sum_{n=0}^{\infty} U_{n}\!\left(x\right) {z}^{n} = \frac{1}{1 - 2 x z + {z}^{2}}
27b2bb
∑n=1∞Tn ⁣(x)znn=−12log⁡ ⁣(1−2xz+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=0∞Tn ⁣(x)znn!=ezxcosh⁡ ⁣(zx2−1)\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=0∞Un ⁣(x)znn!=ezx(cosh⁡ ⁣(zx2−1)+zxsinc⁡ ⁣(izx2−1))\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 ⁣(i∣z∣)∣\left|T_{n}\!\left(z\right)\right| \le \left|T_{n}\!\left(i \left|z\right|\right)\right|
0b3fd6
∣Un ⁣(z)∣≤∣Un ⁣(i∣z∣)∣\left|U_{n}\!\left(z\right)\right| \le \left|U_{n}\!\left(i \left|z\right|\right)\right|
443759
∣Tn ⁣(z)∣≤(∣z∣+∣z∣2+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∣+∣z∣2+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.

2019-06-18 07:49:59.356594 UTC