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In the Sturm-Liouville Boundary Value Problem, there is a special case called Chebyshev's Differential Equation which is defined as:

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:

where

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:

where

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.