Chebyshev polynomial

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


where x=cost.

It is an example of a trigonometric polynomial.

This can be seen to be a polynomialMathworldPlanetmath 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))3-3cos(t)

So we have

T0(x) = 1
T1(x) = x
T2(x) = 2x2-1
T3(x) = 4x3-3x

These polynomials obey the recurrence relation:

Tn+1(x)= 2xTn(x)-Tn-1(x)

for n=1, 2,

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


which can similarly be seen to be polynomials through either a similar process as the above or by the relation Un-1(t)=nTn(t).

The first few are:

U0(x) = 1
U1(x) = 2x
U2(x) = 4x2-1
U3(x) = 8x3-4x

The same recurrence relation also holds for U:

Un+1(x)= 2xUn(x)-Un-1(x)

for n=1, 2,.

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 kindDlmfMathworld