Chebyshev polynomials of the first kind are the trigonometric polynomials defined by:
![](/content/5048/eqn107.gif)
whence
![](/content/5048/eqn108.gif)
For Chebyshev polynomials, a generalized Rodrigues formula is valid
![](/content/5048/eqn109.gif)
A recurrent relationship holds for Chebyshev polynomials
![](/content/5048/eqn110.gif)
Chebyshev polynomials for a negative value of n are defined by the relationship:
![](/content/5048/eqn111.gif)
Chebyshev polynomials of the first kind are orthogonal with respect to a weight function
on the interval [-1, 1]. The orthogonality relationship is:
![](/content/5048/eqn113.gif)
The roots of the polynomial T(x), defined by the equality
, k = 1, 2, ..., n are often used as cusps of quadrature and interpolation formulas.
Chebyshev polynomials of the first kind with a unit coefficient of the higher term, i.e.,
are the polynomials least deviated from zero on the interval [-1, 1], i.e., for any other polynomial Fn(x) of degree n with unit heading coefficient the following relationship holds:
![](/content/5048/eqn116.gif)
![](/content/5048/eqn117.gif)
This property of Chebyshev polynomials is used for constructing optimal iteration algorithms in solving problems of heat transfer with the help of numerical methods.