# Chebyshev polynomials

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##### 1: 18.3 Definitions
###### §18.3 Definitions
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). …
##### 2: 18.41 Tables
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 $H_{n}\left(x\right)$. …
##### 3: 18.7 Interrelations and Limit Relations
###### Chebyshev, Ultraspherical, and Jacobi
18.7.17 $U_{2n}\left(x\right)=W_{n}\left(2x^{2}-1\right),$
18.7.18 $T_{2n+1}\left(x\right)=xV_{n}\left(2x^{2}-1\right).$
##### 4: 18.9 Recurrence Relations and Derivatives
18.9.12 $T_{n+1}\left(x\right)+T_{n}\left(x\right)=(1+x)V_{n}\left(x\right).$
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. …
##### 5: 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: …
$C_{n}\left(x\right)=2T_{n}\left(\tfrac{1}{2}x\right),$
$S_{n}\left(x\right)=U_{n}\left(\tfrac{1}{2}x\right).$
##### 7: 18.5 Explicit Representations
$T_{0}\left(x\right)=1,$
$T_{1}\left(x\right)=x,$
$T_{2}\left(x\right)=2x^{2}-1,$
$T_{3}\left(x\right)=4x^{3}-3x,$
$U_{0}\left(x\right)=1,$
##### 8: 29.15 Fourier Series and Chebyshev Series
###### §29.15(ii) Chebyshev Series
The Chebyshev polynomial $T$ of the first kind (§18.3) satisfies $\cos\left(p\phi\right)=T_{p}\left(\cos\phi\right)$. …
29.15.43 $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p% }T_{2p}\left(\operatorname{sn}\left(z,k\right)\right).$
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 $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)=\sum_{p=0}^{n}A_{2p+1}T_{2p+1}\left% (\operatorname{sn}\left(z,k\right)\right),$
##### 9: 3.11 Approximation Techniques
The Chebyshev polynomials $T_{n}$ are given by …
3.11.7 $T_{n+1}\left(x\right)-2xT_{n}\left(x\right)+T_{n-1}\left(x\right)=0,$ $n=1,2,\dots$,
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. …