After changing $a$ to $2a+1$ and $b$ to $a$ in the following expression $$\operatorname{F}\left(a,b,2b;x\right)=\left(1-\frac{x}{2}\right)^{-a}\operatorname{F}\left(\frac{a}{2},\frac{a}{2}+1,b+\frac{1}{2};\frac{x^2}{\left(2-x\right)^2}\right)$$
how do I show that its a particular case of
$$\operatorname{F}\left(a,b,c;x\right)=x^{1-c}\frac{d^k}{dx^k}\left[\frac{x^{c+k-1}}{\left(c\right)_{k}}\operatorname{F}\left(a,b,c+k;x\right)\right],\ \ \ \ k=0,1,\dots $$
I tried using the identity $$\operatorname{F}\left(a,a+\frac{1}{2},c;\frac{x^2}{\left(2-x\right)^2}\right)=\left(1-\frac{x}{2}\right)^{2a}\operatorname{F}\left(c-\frac{1}{2},2a,2c-1;x\right)$$ This reduced the given expression to $$\operatorname{F}\left(a,2a+1,2a;x\right)$$ please provide any hint to proceed further.Thanks
