Chebyshev polynomials

Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted $T_n(x)$. They are used as an approximation to a least squares fit, and are a special case of the ultra-spherical polynomial (Gegenbauer polynomial) with $\alpha =0$. Chebyshev polynomial of the first kind, $T_n(z)$ can be defined by the contour integral

$T_n(z)=\frac{1}{4\pi i}\oint \frac{(1-t^2)t^{-n-1}}{(1-2tz+t^2)}\mathrm{d}t$

The first seven Chebyshev polynomials of the first kind are:

$T_0(x)=1$

$T_1(x)=x$

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

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

$T_4(x)=8x^4-8x^2+1$

$T_5(x)=16x^5-20x^3+5x$

$T_6(x)=32x^6-48x^4+18x^2-1$

Orthogonality

The Chebyshev polynomials are orthogonal polynomials with respect to the weighting function $(1-x^2{)}^{-1/2}$ such that

$\underset{-1}{\overset{1}{\int }}\frac{T_m(x)T_n(x)}{\sqrt{1-x^2}}\mathrm{d}x=\{\begin{array}{ccc}\frac{1}{2}\pi {\delta }_{(mn)}& ;& m\ne 0,n\ne 0\\ \pi & ;& m=n=0\end{array}$

where ${\delta }_{(mn)}$ is the Kronecker delta.

Applications in statistical mechanics

• Computational implementation of integral equations

See also

• Chebyshev Polynomial of the First Kind -- from Wolfram MathWorld