Editing Classical orthogonal polynomials (section)

=== Chebyshev polynomials ===
The differential equation is

:<math>(1-x^2)\,y'' - x\,y' + \lambda \,y = 0\qquad \text{with}\qquad\lambda = n^2.</math>

This is '''[[Chebyshev equation|Chebyshev's equation]]'''.

The recurrence relation is

:<math>T_{n+1}(x) = 2x\,T_n(x) - T_{n-1}(x).</math>

Rodrigues' formula is

:<math>T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \  \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right).</math>

These polynomials have the property that, in the interval of orthogonality,

:<math>T_n(x) = \cos(n\,\arccos(x)).</math>

(To prove it, use the recurrence formula.)

This means that all their local minima and maxima have values of &minus;1 and +1, that is, the polynomials are "level".  Because of this, expansion of functions in terms of Chebyshev polynomials is sometimes used for [[approximation theory|polynomial approximations]] in computer math libraries.

Some authors use versions of these polynomials that have been shifted so that the interval of orthogonality is [0,&nbsp;1] or [&minus;2,&nbsp;2].

There are also '''Chebyshev polynomials of the second kind''', denoted <math>U_n</math>

We have:

:<math>U_n = \frac{1}{n+1}\,T_{n+1}'.</math>

For further details, including the expressions for the first few
polynomials, see [[Chebyshev polynomials]].