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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.T_{0}(x)\right.=1}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.T_{1}(x)\right.=x}

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.T_{3}(x)\right.=4x^{3}-3x}

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.T_{5}(x)\right.=16x^{5}-20x^{3}+5x}

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

Applications in statistical mechanics

• Computational implementation of integral equations

See also

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