Chebyshev polynomials

For people who check facts

The Chebyshev polynomials named after Pafnuty Chebyshev (Пафнутий Чебышёв), compose a polynomial sequence, and are defined by

for n = 0, 1, 2, 3, .... . That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of De Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even and thus replaceable via the identity cos²(x) + sin²(x) = 1.

Alternatively they can be defined via the recurrence relation

These polynomials are orthogonal with respect to the weight

on the interval [−1,1], i.e., we have

This is because (letting x = cos θ)

The first few polynomials are:

T₀(x)=1

T₁(x)=x

T₂(x)=2x²−1

T₃(x)=4x³−3x

T₄(x)=8x⁴−8x²+1

T₅(x)=16x⁵−20x³+5x

T₆(x)=32x⁶−48x⁴+18x²−1

T₇(x)=64x⁷−112x⁵+56x³−7x

T₈(x)=128x⁸−256x⁶+160x⁴−32x²+1

T₉(x)=256x⁹−576x⁷+432x⁵−120x³+9x