Hermite Polynomials (Graph)
Calculate the table for the Hermite polynomial Hn(x) and display it as a graph.
| \(\normalsize Hermite\ polynomial\ H_n(x)\\ (1)\ y''-2xy'+ny=0,\hspace{20px}y=H_n(x)\\ (2)\ {\large e^{2xt-t^2}}={\large\displaystyle \sum_{n=0}^{\infty}\frac{H_n(x)}{n!}}t^n\\ (3)\hspace{5px}{\large\int_{\tiny -\infty}^{\tiny \infty}}H_n(x)H_m(x){\large e^{-x^2}}dx={\large 2^nn!\sqrt{n}\delta_{mn}}\\ (4)\ H_n(x)\\ \hspace{10px}=2^n{\large\sqrt{\pi}}\Biggl({\large\frac{{}_{\tiny 1}F_{\tiny 1} (-\frac{n}{2};\frac{1}{2};x^2)}{\Gamma(\frac{1-n}{2})}}-{\large\frac{{}_{\tiny 1}F_{\tiny 1} (\frac{1-n}{2};\frac{3}{2};x^2)2x}{\Gamma(-\frac{n}{2})}}\Biggr)\\\) |
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