Type II Chebyshev Polynomials (Graph)
Calculate and graph the table of the second-kind Chebyshev polynomials Un(x).
| \(\normalsize Chebyshev\ polynomial\\ \hspace{120px} of\ the\ 2nd\ kind\ U_n(x)\\ (1)\ (1-x^2)y''-xy'+n^2y=0\\ \hspace{25px} y=U_n(x)\\ (2)\ {\large\frac {1}{1-2xt+t^2}}={\large\displaystyle \sum_{n=0}^{\infty}} U_n(x)t^n\\ (3)\ {\large\int_{\tiny -1}^{\tiny 1}\sqrt{1-x^2}}U_n(x)U_m(x)dx={\large\frac{\pi}{2}}\delta_{mn}\\ (4)\ U_n(x)=(n+1){}_{\small 2}F_{\small 1} (-n,n+2;\ {\large\frac{3}{2}};{\large\frac{1-x}{2}})\\ \hspace{70px}={\large\frac{\sin((n+1) \cos^{-1}x)}{\sin( \cos^{-1}x)}}\\\) |
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