In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are two sequences of orthogonal polynomials which are related to de Moivre's formula. These polynomials are also known for their elegant Trigonometric properties, and can also be defined recursively. They are very helpful in Trigonometry, Complex Analysis, and other branches of Algebra.
The Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula.
There are two kinds of these polynomials. The first kind $T_n$ is defined by the recurrence $$\begin{align} T_0(x)&=1\\ T_1(x)&=x\\ T_{n+1}(x)&=2xT_n(x)-T_{n-1}(x) \end{align}$$ The second kind $U_n$ is defined by the same recurrence, but with $U_1(x)=2x$.
These polynomials also satisfy the trigonometric identities $$T_n(\cos\theta)=\cos(n\theta)\qquad U_n(\cos\theta)\sin\theta=\sin(n+1)\theta.$$
