Writing an integral in terms of the Hypergeometric Function

Question

I have the following function defined as an integral:

$G(x,k,s) = 1 - (k-1) \int_0^{x/s} (1-t)^{k-2} dt$

Or alternatively directly as:

$G(x,k,p) = (1-\frac{x}{s})^{k-1}$

Is there any way (or approximation) to write it as a function of the Gaussian hypergeometric function $_2F_1(a,b;c,1)$ or $_2F_1(a,b;c,z)$?

Answer 1

$$_2F_1(a,b;c;z)=\frac{\Gamma(c) }{\Gamma(b)\Gamma(c-b)}\int_0^1\theta^{b-1}(1-\theta)^{c-b-1}(1-\theta z)^{-a}d\theta$$ Let $\quad a=2-k \quad;\quad b=1\quad;\quad c=2 $ $$_2F_1(2-k,1;2;z)=\int_0^1(1-\theta z)^{k-2}d\theta$$ Let $\quad \theta=\frac{s}{x}t\quad;\quad z=\frac{x}{s}$

$_2F_1(2-k,1;2;x/s)=\int_0^{x/s}(1-t)^{k-2}\frac{s}{x}dt$ $$\int_0^{x/s}(1-t)^{k-2}dt = \frac{x}{s}\:_2F_1\left((2-k)\:,\:1\:;\:2\:;\:\frac{x}{s}\right)$$