# Gegenbauer polynomial (chart) Calculator

## Calculates a table of the Gegenbauer polynomial Cnλ(x) and draws the chart.

 n n=0,1,2,... λ [ initial value x -1≦x≦1 increment repetition ]
 $\normal Gegenbauer\ polinomial\ C_n^\lambda(x)\\[10](1)\ (1-x^2)y''-(2\lambda+1)xy'+n(n+2\lambda)y=0\\\hspace{25}y=C_n^\lambda(x)\\(2)\ {\large\frac {1}{(1-2xt+t^2)^\lambda}}={\large\sum_{\small n=0}^{\small\infty}} C_n^\lambda(x)t^n\\(3)\hspace{5}{\large\int_{\tiny -1}^{\hspace{25}\tiny 1}}(1-x^2)^{\lambda-\frac{1}{2}}C_n^\lambda(x)C_m^\lambda(x)dx\\\hspace{110}={\large\frac{\pi\Gamma(n+2\lambda)}{2^{2\lambda-1}(n+\lambda)n!\Gamma^2(\lambda)}}\delta_{mn}\\[10](4)\hspace{2} C_n^\lambda(x)={\normal\frac{\Gamma(n+2\lambda)}{n!\Gamma(2\lambda)}}{}_{\small 2}F_{\small 1} ({\small-}n,n+2\lambda;\lambda+\frac{1}{2};\frac{1-x}{2})\\$

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