An “Inverse Gauss Hypergeometric function” with respect to $z$ in terms of a differential equation would define many special case inverse functions. Define:
$$\,_2\text F_1(a,b;c;z)=\sum_{n=0}^\infty \frac{(a)_n(b)_n z^n}{(c)_nn!}$$
and Confluent Hypergeometric function
$$\,_1\text F_1(a;b;z)= \sum_{n=0}^\infty \frac{(a)_nz^n}{(b)_nn!}=\lim_{p\to\infty} \,_2\text F_1\left(a,p;b;\frac zp\right) $$
with the Pochhammer Symbol
Example 1: The Lower Incomplete Gamma function:
$$\gamma(a,z)=\frac{z^a}{a}\,_1\text F_1(a;a+1;-z)=w(a,z):zw’’+(z-a+1)w’=0$$
therefore the differential equation of the inverse for all values of $a,z$ with respect to $z$ is:
$$\gamma(a,y)=z\implies y=y(a,z): yy’’-y’^2(y-a+1)=0$$
Example 2:
$$\text B_z(a,b)=\frac{z^a}a\,_2\text F_1(a,1-b;a+1;z)=w(z,a,b):(1-z)zw’’+((a+b-2)z-a+1)w’=0$$
therefore the differential equation of the inverse for all values of $a,b,z$ with respect to $z$ is:
$$\text B_y(a,b)=z\implies y=y(z,a,b):(1-y)yy’’-((a+b-2)y-a+1)y’^2=0$$
The goal differential equation:
The difference in the differential equations of $y$ and $w$ seems to be changing all $z$s into $y$s, but this is not the method for finding the differential equation of an inverse function. Here is a particular solution of the Hypergeometric Differential Equation
$$\,_2\text F_1(a,b;c;z)=w(a,b,c,z): (1-z)zw’’+(c-(a+b+1)z)w’-abw=0$$
and the inverse function is:
$$\,_2\text F_1(a,b;c;y) =z\implies y=y(a,b,c,z):y’=\frac c{ab \,_2\text F_1(a+1,b+1;c+1;y)} $$
Is there a way to simplify the above differential equation into a form similar to the Hypergeometric Differential equation form?
Please correct me and give me feedback!