Chebyshev polynomial

The Chebyshev polynomials of first kind are defined by the simple formula

$$T_n(x)=\cos (nt),$$

where $x=\cos t$.

It is an example of a trigonometric polynomial.

This can be seen to be a polynomial by expressing $\cos (kt)$ as a polynomial of $\cos (t)$, by using the formula for cosine of angle-sum:

$\cos (1t)$ $=$ $\cos (t)$

$\cos (2t)$ $=$ $\cos (t)\cos (t)-\sin (t)\sin (t)=2{(\cos (t))}^2-1$

$\cos (3t)$ $=$ $4{(\cos (t))}^3-3\cos (t)$

$\mathrm{⋮}$

So we have

$T_0(x)$ $=$ $1$

$T_1(x)$ $=$ $x$

$T_2(x)$ $=$ $2x^2-1$

$T_3(x)$ $=$ $4x^3-3x$

$\mathrm{⋮}$

These polynomials obey the recurrence relation:

$$T_{n+1}(x)=\mathrm{\hspace{0.33em}2}x{T}_n(x)-T_{n-1}(x)$$

for $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots }$

Related are the Chebyshev polynomials of the second kind that are defined as

$$U_{n-1}(\cos t)=\frac{\sin (nt)}{\sin (t)},$$

which can similarly be seen to be polynomials through either a similar process as the above or by the relation $U_{n-1}(t)=n{T}_n^{\prime }(t)$.

The first few are:

$U_0(x)$ $=$ $1$

$U_1(x)$ $=$ $2x$

$U_2(x)$ $=$ $4x^2-1$

$U_3(x)$ $=$ $8x^3-4x$

$\mathrm{⋮}$

The same recurrence relation also holds for $U$:

$$U_{n+1}(x)=\mathrm{\hspace{0.33em}2}x{U}_n(x)-U_{n-1}(x)$$

for $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots }$.

Title	Chebyshev polynomial
Canonical name	ChebyshevPolynomial
Date of creation	2013-03-22 12:22:56
Last modified on	2013-03-22 12:22:56
Owner	drini (3)
Last modified by	drini (3)
Numerical id	11
Author	drini (3)
Entry type	Definition
Classification	msc 42C05
Classification	msc 42A05
Classification	msc 33C45
Related topic	Polynomial
Defines	Chebyshev polynomial of first kind
Defines	Chebyshev polynomial of second kind