# 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:

 $\displaystyle\cos(1t)$ $\displaystyle=$ $\displaystyle\cos(t)$ $\displaystyle\cos(2t)$ $\displaystyle=$ $\displaystyle\cos(t)\cos(t)-\sin(t)\sin(t)=2(\cos(t))^{2}-1$ $\displaystyle\cos(3t)$ $\displaystyle=$ $\displaystyle 4(\cos(t))^{3}-3\cos(t)$ $\displaystyle\vdots$

So we have

 $\displaystyle T_{0}(x)$ $\displaystyle=$ $\displaystyle 1$ $\displaystyle T_{1}(x)$ $\displaystyle=$ $\displaystyle x$ $\displaystyle T_{2}(x)$ $\displaystyle=$ $\displaystyle 2x^{2}-1$ $\displaystyle T_{3}(x)$ $\displaystyle=$ $\displaystyle 4x^{3}-3x$ $\displaystyle\vdots$

These polynomials obey the recurrence relation:

 $T_{n+1}(x)\;=\;2xT_{n}(x)-T_{n-1}(x)$

for $n=1,\,2,\,\ldots$

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 $U_{n-1}(t)=nT_{n}^{\prime}(t)$.

The first few are:

 $\displaystyle U_{0}(x)$ $\displaystyle=$ $\displaystyle 1$ $\displaystyle U_{1}(x)$ $\displaystyle=$ $\displaystyle 2x$ $\displaystyle U_{2}(x)$ $\displaystyle=$ $\displaystyle 4x^{2}-1$ $\displaystyle U_{3}(x)$ $\displaystyle=$ $\displaystyle 8x^{3}-4x$ $\displaystyle\vdots$

The same recurrence relation also holds for $U$:

 $U_{n+1}(x)\;=\;2xU_{n}(x)-U_{n-1}(x)$

for $n=1,\,2,\,\ldots$.

Title Chebyshev polynomial ChebyshevPolynomial 2013-03-22 12:22:56 2013-03-22 12:22:56 drini (3) drini (3) 11 drini (3) Definition msc 42C05 msc 42A05 msc 33C45 Polynomial Chebyshev polynomial of first kind Chebyshev polynomial of second kind