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.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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 T_n (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)y^” – xy + n²y = 0

and the recursion formula

T_n+1 (x) = 2xT_n(x) – T_{n – 1}(x)

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

Chebyshev polynomials of the second kind U_n (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)U_{n – 1} (x) = xT_n (x) – T_n+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.)

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