> [!NOTE] Definition (Chebyshev polynomials of the first kind) >The [[Ring of Polynomial Forms|polynomials]] $T$ in indeterminate $x$ that satisfy the [[Recurrence Relation|recurrence relation]] $T_{n+1}(x) = 2xT_{n}(x)-T_{n-1}(x)$with $T_{0}(x)=1$ and $T_{1}(x)=x$ are known as Chebyshev polynomials of the first kind. > # Properties > [!NOTE] Theorem (Trigonometric Definition of Chebyshev polynomials of the first kind) >We may define $T_{n}$ as the polynomial that satisfies $T_{n}(\cos\theta)=\cos n\theta.$ >Proof. Using [[Trigonometric Functions#^df0e36|addition formula]], for any $n$ $\cos (n+1) \theta + \cos (n-1) \theta = 2 \cos \theta \cdot \cos n\theta$Hence indeed $T_{n+1}(x)=2xT_{n}(x)-T_{n-1}(x).$ Proposition (Closed formula for Chebyshev polynomials of the first kind)