Chebyshev Polynomials

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 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.)