# 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 $T_{N+1}\left(x\right)$: … Note that in this reference the definitions of the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ and $W_{n}\left(x\right)$ are the converse of the definitions in this chapter. 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: 18.1 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: $T^{*}_{n}\left(x\right)$, $U^{*}_{n}\left(x\right)$.

• $C_{n}\left(x\right)=2T_{n}\left(\tfrac{1}{2}x\right),$
$S_{n}\left(x\right)=U_{n}\left(\tfrac{1}{2}x\right).$
In Mason and Handscomb (2003), the definitions of the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ and $W_{n}\left(x\right)$ are the converse of the definitions in this chapter.
##### 4: 18.9 Recurrence Relations and Derivatives
18.9.10 $(1-x^{2})U_{n}\left(x\right)=-\tfrac{1}{2}\left(T_{n+2}\left(x\right)-T_{n}% \left(x\right)\right).$
18.9.11 $W_{n}\left(x\right)+W_{n-1}\left(x\right)=2T_{n}\left(x\right),$
##### 6: 18.5 Explicit Representations
###### Chebyshev
$T_{0}\left(x\right)=1,$
$T_{1}\left(x\right)=x,$
$U_{0}\left(x\right)=1,$
$U_{1}\left(x\right)=2x,$
##### 8: 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. …
##### 10: 16.26 Approximations
For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).