# Chebyshev Polynomials

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## Chebyshev polynomials

[′cheb·ə·shəf ‚päl·i′nō·mē·əlz]*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 It might be outdated or ideologically biased.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, *T*_{0} = 1, *T*_{1} = *x*, *T*_{2} = 2*x*^{2} – 1, *T*_{3} = 4*x*^{3} – 3*x* and *T*_{4} = 8*x*^{4} – 8*x*^{2} + 1.

The polynomials *T*_{n} (*x*) are orthogonal with respect to the weight function (1 – *x*^{2})^{–½} on the interval [–1, +1] (*see*ORTHOGONAL POLYNOMIALS). They satisfy the differential equation

(1 – *x*^{2})*y*^{”} – *xy* + *n*^{2}*y* = 0

and the recursion formula

*T*_{n+1} (*x*) = 2*xT*_{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 –

*x*

^{2})

^{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 – *x*^{2})*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.)