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

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

The first seven Chebyshev polynomials of the first kind are:

$\left.T_{0}(x)\right.=1$

$\left.T_{1}(x)\right.=x$

$\left.T_{2}(x)\right.=2x^{2}-1$

$\left.T_{3}(x)\right.=4x^{3}-3x$

$\left.T_{4}(x)\right.=8x^{4}-8x^{2}+1$

$\left.T_{5}(x)\right.=16x^{5}-20x^{3}+5x$

$\left.T_{6}(x)\right.=32x^{6}-48x^{4}+18x^{2}-1$

Orthogonality

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

$\int _{-1}^{1}{\frac {T_{m}(x)T_{n}(x)}{\sqrt {1-x^{2}}}}\mathrm {d} x=\left\{{\begin{array}{lll}{\frac {1}{2}}\pi \delta _{(mn)}&;&m\neq 0,n\neq 0\\\pi &;&m=n=0\end{array}}\right.$

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