The Hypergeometric Function

The Gaussian Hypergeometric Equation

Series Representation (in Pochhammer notation)

`""_2F_1(e,f;g;z)=sum_(n=0)^(oo) (e^(bar n)f^(bar n))/(g^(bar n))z^n/(n!)`

Series Representation (in Gamma notation)

`""_2F_1(e,f;g;z)= (Gamma(g))/(Gamma(e)Gamma(f)) sum_(n=0)^(oo) (Gamma(e+n)Gamma(f+n))/(Gamma(g+n)) z^n/(Gamma(n+1))`

The Euler Integral

`""_2F_1(e,f;g;z)=(Gamma(g))/(Gamma(f)Gamma(g-f)) int_0^1 ((1-u)^(g-f-1)u^(f-1))/(1-uz)^edu`

Differential Equation

`z(1-z)(d^2w)/(dz^2)+[c-(a+b+1)z](dw)/(dz)-abw=0`