How to transform this MeijerG $G_{3,3}^{2,3}$ into a hypergeometric function $ _2F_1 $

Question

I want to transform the MeijerG function $G_{3,3}^{2,3}$ below into a hypergeometric function $ _2F_1 $:

$$G_{3,3}^{2,3}\left(\frac{\kappa -z}{z},\frac{1}{2} | \begin{array}{c} \frac{1-\kappa }{2},2-\frac{\kappa }{2},-\frac{\kappa }{2} \\ 0,\frac{1}{2},\frac{1-\kappa }{2} \\ \end{array} \right)$$

MeijerG[{{(1 - \[Kappa])/2, 2 - \[Kappa]/2, -(\[Kappa]/2)}, {}}, {{0, 1/2}, {(1 - \[Kappa])/2}}, (-z + \[Kappa])/z, 1/2]

According to @yarchik's answer, I applied FunctionExpand, I find:

FunctionExpand[MeijerG[{{(1 - \[Kappa])/2,2 - \[Kappa]/2, -(\[Kappa]/2)}, {}}, {{0, 1/2}, {(1 - \[Kappa])/2}}, (-z + \[Kappa])/z, 1/2]] // FullSimplify 

(* Sqrt[\[Pi]] (Gamma[-1 + \[Kappa]/2] Gamma[     1 + \[Kappa]/2] Hypergeometric2F1[1/2 (-2 + \[Kappa]), (2 + \[Kappa])/2, 1/2, (z - \[Kappa])^2/z^2] + (2 Gamma[1/2 (-1 + \[Kappa])] Gamma[(3 + \[Kappa])/
 2] (\[Kappa] (-2 z + \[Kappa]) Hypergeometric2F1[1/2 (-1 + \[Kappa]), (3 + \[Kappa])/2, -(1/2), (z - \[Kappa])^2/
    z^2] - (-4 z \[Kappa] (1 + \[Kappa]) + 2 \[Kappa]^2 (1 + \[Kappa]) + z^2 (1 + 2 \[Kappa])) Hypergeometric2F1[1/2 (-1 + \[Kappa]), (3 + \[Kappa])/2, 1/2, (z - \[Kappa])^2/z^2]))/(z (z - \[Kappa]) (-4 + \[Kappa]^2)))  *) 

Now how to transform this combination of $ _2F_1 $ functions into a single $ _2F_1 $ function?

Answer 1

One possibility to get a compact result is

y = y = MeijerG[{{(1 - k)/2, 2 - k/2, -(k/2)}, {}}, {{0, 1/2}, {(1 - k)/2}}, (k - z)/z, 1/2]
FunctionExpand[y] // FullSimplify

$$\frac{y}{\sqrt{\pi}}=g_1+\frac{2\kappa (\kappa -2 z)}{\left(\kappa ^2-4\right) z (z-\kappa )}g_2 -\frac{2 \left(2 \kappa ^2 (\kappa +1)+(2 \kappa +1) z^2-4 \kappa (\kappa +1) z\right)}{\left(\kappa ^2-4\right) z (z-\kappa )}g_3,\\ g_1=\Gamma \left(\frac{\kappa -2 }{2}\right) \Gamma \left(\frac{\kappa +1 }{2}\right)\, _2F_1\left(\frac{\kappa -2}{2},\frac{\kappa +2}{2};\frac{1}{2};u\right),\\ g_2=\Gamma \left(\frac{\kappa -1}{2}\right) \Gamma \left(\frac{\kappa +3}{2}\right)\, _2F_1\left(\frac{\kappa -1}{2},\frac{\kappa +3}{2};-\frac{1}{2};u\right),\\ g_3=\Gamma \left(\frac{\kappa -1}{2}\right) \Gamma \left(\frac{\kappa +3}{2}\right)\, _2F_1\left(\frac{\kappa -1}{2},\frac{\kappa +3}{2};\frac{1}{2};u\right),\\ u=\frac{(z-\kappa )^2}{z^2}. $$