Chebyshev U
The Chebyshev polynomials of the second kind are orthogonal polynomials defined by $$U_n(x) = \sin(n \mathrm{arcsin}(x)).$$
Properties
- REDIRECT Chebyshev differential equation
Theorem: The following formula holds: $$\int_{-1}^1 \dfrac{U_m(x)U_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0\\ 0 &; m=n=0. \end{array} \right.$$
Proof: █
Theorem
The following formula holds for $n \in \{0,1,2,\ldots\}$: $$U_n(x) = (n+1){}_2F_1 \left( -n,n+2 ; \dfrac{3}{2}; \dfrac{1-x}{2} \right),$$ where $U_n$ denotes a Chebyshev polynomial of the second kind and ${}_2F_1$ denotes hypergeometric 2F1.
Proof
References
Theorem
The following formula holds for $n \in \{1,2,3,\ldots\}$: $$U_n(x)=\sqrt{1-x^2}C_{n-1}^1(x),$$ where $U_n$ denotes a Chebyshev polynomial of the second kind and $C_{n-1}^1$ denotes a Gegenbauer C polynomial.
