Chebyshev U

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The Chebyshev polynomials of the second kind are orthogonal polynomials defined by $$U_n(x) = \sin(n \mathrm{arcsin}(x)).$$

Properties[edit]

Theorem: The Chebyshev T and Chebyshev U polynomials $T_n(x)$ and $U_n(x)$ are two independent solutions of the following equation, called Chebyshev's equation: $$(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+n^2y=0.$$

Proof:

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: $$U_n(x) = (n+1){}_2F_1 \left( -n,n+2 ; \dfrac{3}{2}; \dfrac{1-x}{2} \right),$$ where $U_n$ denotes a Chebyshev U polynomial and ${}_2F_1$ denotes the hypergeometric pFq.

Proof:

Theorem: The following formula holds: $$U_n(x)=\sqrt{1-x^2}C_{n-1}^1(x),$$ where $U_n$ denotes a Chebyshev U polynomial and $C_{n-1}^1$ denotes a Gegenbauer C polynomial.

Proof:

Orthogonal polynomials