Chebyshev polynomials of the first kind are the trigonometric polynomials defined by:

whence

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 F_{n}(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.

Heat & Mass Transfer, and Fluids Engineering