Chebyshev polynomials

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.

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
T₁₀(x)=512x¹⁰-1280x⁸+1120x⁶-400x⁴+50x²-1
T₁₁(x)=1024x¹¹-2816x⁹+2816x⁷-1232x⁵+220x³-11x
T₁₂(x)=2048x¹²-6144x¹⁰+6912x⁸-3584x⁶+840x⁴-72x²+1
T₁₃(x)=4096x¹³-13312x¹¹+16640x⁹-9984x⁷+2912x⁵-364x³+13x
T₁₄(x)=8192x¹⁴-28672x¹²+39424x¹⁰-26880x⁸+9408x⁶-1568x⁴+98x²-1
T₁₅(x)=16384x¹⁵-61440x¹³+92160x¹¹-70400x⁹+28800x⁷-6048x⁵+560x³-15x
T₁₆(x)=32768x¹⁶-131072x¹⁴+212992x¹²-180224x¹⁰+84480x⁸-21504x⁶+2688x⁴-128x²+1