Orthogonal Polynomials
  Chebyshev Plot

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 .

Important Properties

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 n.

Based on ,

• Pn(x) is an even function, when n is even.

• Pn(x) is an odd function, when n is odd.

Recurrence Relation: A Chebyshev Polynomial at one point can be expressed by neighboring Chebyshev Polynomials at the same point.

Special Results

Chebyshev Polynomial Related Calculator

where , , are Chebyshev Polynomials of the second kind. A linear combination of Chebyshev Polynomials of the First (Tn) and Second (Un) Kinds forms the general solution of the Chebyshev Equation.