# Gaussian hypergeometric function Calculator

## Calculates Gaussian hypergeometric function 2F1(a,b;c;z).

 $\normal Hypergeometric\ function\ of\ the\ 1st\ kind\\[15]\large {}_2F_1(a,b;c;z)=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^2}{2!}+\cdots\\\hspace{90}=\sum_{\small n=0}^{\small \infty}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}\\[10]$
 a b c z 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt 2F1(a,b;c;z)
 $\normal Hypergeometric\ differential\ equation\\[15](1)\ z(1-z)y''+(c-(a+b+1)z))y'-aby=0\\\hspace{25} y={}_2F_1(a,b;c;z)\\[10](2)\ {}_2F_1(a,b;c;z)=1+{\large\frac{ab}{c}}z+{\large\frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^2}{2!}}+\cdots\\\hspace{90}={\large \sum_{\small n=0}^{\small \infty}}{\large\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}}\\[10]$

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