Chebyshev polynomials of the first kind are the trigonometric polynomials defined by:
For Chebyshev polynomials, a generalized Rodrigues formula is valid
A recurrent relationship holds for Chebyshev polynomials
Chebyshev polynomials for a negative value of n are defined by the relationship:
Chebyshev polynomials of the first kind are orthogonal with respect to a weight function on the interval [-1, 1]. The orthogonality relationship is:
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:
This property of Chebyshev polynomials is used for constructing optimal iteration algorithms in solving problems of heat transfer with the help of numerical methods.