Chebyshev polynomials of the first kind are orthogonal polynomials defined for $n=0,1,2,\ldots$ by $$T_n(x) = \cos(n \mathrm{arccos}(x)),$$ where $\cos$ denotes cosine and $\mathrm{arccos}$ denotes arccos.

Properties

T_(n+1)(x)-2xT_n(x)+T_(n-1)(x)=0

Theorem (Orthogonality): The following formula holds: $$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0 \\ \pi &; m=n=0. \end{array} \right.$$

Proof: █

Theorem

The following formula holds for $n \in \{0,1,2,\ldots\}$: $$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and ${}_2F_1$ denotes the hypergeometric pFq.

Proof

References

Theorem

The following formula holds for $n \in \{1,2,3,\ldots\}$: $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda},$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and $C_n^{\lambda}$ denotes a Gegenbauer C polynomial.