# Chebyshev Polynomials

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## Chebyshev polynomials

[′cheb·ə·shəf ‚päl·i′nō·mē·əlz]
(mathematics)
A family of orthogonal polynomials which solve Chebyshev's differential equation.

## Chebyshev Polynomials

Chebyshev polynomials of the first kind are a special system of polynomials of successively increasing degree. For n = 0, 1, 2, . . . they are defined by the formula

In particular, T0 = 1, T1 = x, T2 = 2x2 – 1, T3 = 4x3 – 3x and T4 = 8x4 – 8x2 + 1.

The polynomials Tn (x) are orthogonal with respect to the weight function (1 – x2)–½ on the interval [–1, +1] (seeORTHOGONAL POLYNOMIALS). They satisfy the differential equation

(1 – x2)yxy + n2y = 0

and the recursion formula

Tn+1 (x) = 2xTn(x) – Tn – 1(x)

Chebyshev polynomials of the first kind are a special case of the Jacobi polynomials Pn(α,β)(x):

Chebyshev polynomials of the second kind Un (x) are a system of polynomials that are orthogonal with respect to the weight function (1 – x2)½ on the interval [–1, +1]. The relation between Chebyshev polynomials of the second kind and Chebyshev polynomials of the first kind is given by, for example, the recursion formula

(1 – x2)Un – 1 (x) = xTn (x) – Tn+l(x)

### REFERENCES

Chebyshev, P. L. Poln. sobr. soch., vols. 2–3. Moscow-Leningrad, 1947–48.
Szegö, G. Ortogonal’nye mnogochleny. Moscow, 1962. (Translated from English.)
References in periodicals archive ?
One of the most important properties is that Chebyshev polynomials are the so-called semi-group property which establishes that
m]) in (2) denote the k degree Chebyshev polynomials of the first kind, whereas [phi] represents the instantaneous MF values, and [[PHI]].
Chebyshev polynomials and modified cyclotomic polynomials.
Author Brian George Spencer Doman examines classical orthogonal polynomials and their additional properties, covering hermite polynomials, associated Laguerre polynomials, Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials, associated Legendre functions, Jacobi polynomials, and many other related mathematical subjects over twelve chapters and appendices.
Since the second kinds of Chebyshev Polynomials are orthogonal to each other, the operational matrices based on Chebyshev Polynomials greatly reduce the size of computational work while accurately providing the series solution.
The numerical simulations are done by the Chebyshev collocation method with Chebyshev polynomials used for approximating physical quantities and finding the spatial derivatives.
As an application, two different formulations are derived for Hermite interpolation polynomials in the interval [-1,1] with the zeros of the four families of Chebyshev polynomials as nodes.
They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials.
Cortes, "Direct determination of the coefficients of chebyshev polynomials and other consideration," 35th Midwest Symposium on Circuits and Systems, 466-468, 1992.
Determination of the zeros of a linear combination of Chebyshev polynomials.
When formulating equations stemming from the boundary conditions on the beam's end (s = [- or +]1) one uses the expansions of function (6), formulas (2)-(3) for internal forces and the following formulas for calculating the Chebyshev polynomials in points s = [- or +]1 (Paszkowski 1975):

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