where is a real number. The solutions of this equation are called Chebyshev Functions of degree .
If is a non-negative integer, i.e., , the Chebyshev Functions are often referred to as Chebyshev Polynomials .
Rodrigues' Formula: The Chebyshev Polynomials can be expressed by Rodrigues' formula:
Generating Function: The generating function of a Chebyshev Polynomial is:
Orthogonality: Chebyshev Polynomials , , form a complete orthogonal set on the interval with respect to the weighting function . It can be shown that:
By using this orthogonality, a piecewise continuous function in can be expressed in terms of Chebyshev Polynomials:
This orthogonal series expansion is also known as a Fourier-Chebyshev Series expansion or a Generalized Fourier Series expansion.
Even/Odd Functions: Whether a Chebyshev Polynomial is an even or odd function depends on its degree .
Based on ,
is an even function, when is even.
is an odd function, when is odd.
Recurrence Relation: A Chebyshev Polynomial at one point can be expressed by neighboring Chebyshev Polynomials at the same point.