In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted T_n and Chebyshev polynomials of the second kind which are denoted U_n. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebyshef or Tschebyscheff.

The Chebyshev polynomials T_n or U_n are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.

Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides the best approximation to a continuous function under the maximum norm.

In the study differential equations they arise as the solution to the Chebyshev differential equation

for the polynomials of the first and second kind, respectively. These equation are special cases of the Sturm-Liouville differential equation.

Definition

The Chebyshev polynomials of the first kind are defined by the recurrence relation

One example of a generating function for this recurrence relation is

The Chebyshev polynomials of the second kind are defined by the recurrence relation

One example of a generating function for this recurrence relation is

Examples

This image shows the first few Chebyshev polynomials of the first kind in the domain _1¼<x<1¼, _1¼<y<1¼; the flat T₀, and T₁, T₂, T₃, T₄ and T₅.

The first few Chebyshev polynomials of the first kind are

This image shows the first few Chebyshev polynomials of the second kind in the domain _1¼<x<1¼, _1¼<y<1¼; the flat U₀, and U₁, U₂, U₃, U₄ and U₅. Although not visible in the image, U_n(1)=n+1 and U_n(_1)=(n+1)(_1)ⁿ.

The first few Chebyshev polynomials of the second kind are

Trigonometric definition

The Chebyshev polynomials of the first kind can be defined by the trigonometric identity

for n = 0, 1, 2, 3, .... . That cos(nx) is an nth_degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of De Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even and thus replaceable via the identity cos²(x) + sin²(x) = 1.

Written explicitly

Similarly, the polynomials of the second kind satisfy

Notes

The Chebyshev polynomials of the first and second kind are closely related by the following equations

Both the T_n and the U_n form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight

on the interval [−1,1], i.e., we have

This is because (letting x = cos θ)

Similarly, the polynomials of the second kind are orthogonal with respect to the weight

(which, when normalized to form a probability measure, is the Wigner semicircle distribution). The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials.

Polynomial in Chebyshev form

A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form

where T_n is the nth Chebyshev polynomial.

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.

Chebyshev roots

A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [_1,1]. The roots are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric form one can easily prove that the roots of T_n are

Similarly, the roots of U_n are

See also

• Chebyshev nodes
• Chebyshev filter
• Legendre polynomials
• Hermite polynomials

References

• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Chapter 22. New York: Dover, 1972.