Jacobi Polynomials (Graph)
Calculates and plots a table of the Jacobi polynomials Pnα,β(x).
| \(\normalsize Jacobi\ polynomial\ P_n^{\alpha,\beta}(x)\\ (1)\ (1-x^2)y''-(\beta-\alpha-(\alpha+\beta+2)x)y'\\ \hspace{150px}+n(n+\alpha+\beta+1)y=0\\ \hspace{25px} y=P_n^{\alpha,\beta}(x)\\ (2)\hspace{5px}{\large\int_{\tiny -1}^{\tiny 1}}(1-x)^\alpha(1+x)^\beta P_n^{\alpha,\beta}(x)P_m^{\alpha,\beta}(x)dx\\ \hspace{40px}={\large\frac{2^{\alpha+\beta+1}\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{(2n+\alpha+\beta+1)n!\Gamma(n+\alpha+\beta+1)}}\delta_{mn}\\ (3)\ P_n^{\alpha,\beta}(x)={\large\frac{\Gamma(n+\alpha+1)}{\Gamma(n+1)\Gamma(\alpha+1)}}\\ \hspace{80px}\times{}_{\small 2}F_{\small 1} (-n,n+\alpha+\beta+1;\alpha+1;\frac{1-x}{2})\\\) |
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