# Jacobi polynomial (chart)

## Calculates a table of the respective values of the Jacobi polynomial Pnα,β(x) and draws the chart.

n
 n=0,1,2,...
α
β
[ initial value x
 -1≦x≦1
increment
 repetition ]

x Pnα,β(x)

 $\normal Jacobi\ polinomial\ P_n^{\alpha,\beta}(x)\\[10](1)\ (1-x^2)y''-(\beta-\alpha-(\alpha+\beta+2)x)y'\\\hspace{150}+n(n+\alpha+\beta+1)y=0\\\hspace{25} y=P_n^{\alpha,\beta}(x)\\(2)\hspace{5}{\large\int_{\tiny -1}^{\hspace{25}\tiny 1}}(1-x)^\alpha(1+x)^\beta P_n^{\alpha,\beta}(x)P_m^{\alpha,\beta}(x)dx\\\hspace{40}={\large\frac{\2^{\alpha+\beta+1}\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{(2n+\alpha+\beta+1)n!\Gamma(n+\alpha+\beta+1)}}\delta_{mn}\\[10](3)\ P_n^{\alpha,\beta}(x)={\large\frac{\Gamma(n+\alpha+1)}{\Gamma(n+1)\Gamma(\alpha+1)}}\\\hspace{80}\times{}_{\small 2}F_{\small 1} (-n,n+\alpha+\beta+1;\alpha+1;\frac{1-x}{2})\\$

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