For hypergeometric function $_2F_1(a_1,a_2;b_1;z)$ there exists Euler/Pfaff transformations: $$_2F_1(a_1,a_2;b_1;z)=((1-z)^{b_1-a_1-a_2})_2F_1(b_1-a_1,b_1-a_2;b_1;z),\quad \text{Euler transformation}$$
$$_2F_1(a_1,a_2;b_1;z)=((1-z)^{-a_1})_2F_1\left(a_1,b_1-a_2;b_1;\frac{z}{z-1}\right),\quad \text{Pfaff transformation}$$
$$_2F_1(a_1,a_2;b_1;z)=((1-z)^{-a_2})_2F_1\left(b_1-a_1,a_2;b_1;\frac{z}{z-1}\right),\quad \text{Pfaff transformation}$$
We are looking forsimilar formulas for $_1F_2(a_1;b_1,b_2;z)$, $_2F_3(a_1,a_2;b_1,b_2,b_3;z)$, and $_3F_4(a_1,a_2,a_3;b_1,b_2,b_3,b_4;z)$.
Any comments and references are welcome!
Thanks- mike