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 Definition 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: 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. Special Results   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.     