Question/Motivation
I am trying to classify all of the entire (i.e. holomorphic) solutions to a simple-looking ODE of the form
$$z\frac{\partial^2f}{\partial z^2} + (az+b)\frac{\partial f}{\partial z}+(cz+d)f=0.\,\,\,\,\,\,\,\,\,(1)$$
without directly solving the recursion (i.e. by applying transformations to simplify the ODE into a hypergeometric ODE, and then taking the standard holomorphic solutions). What indicates that this may be a difficult task is the limiting case $a=d=0$:
$$z\frac{\partial^2f}{\partial z^2} + b\frac{\partial f}{\partial z}+czf=0.$$ $$$$
Mathematica/Wikipedia gets stuck:
In this limiting case, the standard derivation in Wikipedia, in addition to Mathematica's DSolve[], both seem to miss a holomorphic solution. In particular, the Wikipedia derivation and Mathematica both spit out a set of solutions of the form \begin{align*} f_1&=\,_0F_1(b;-cz^2/4)\\ f_2&=(-cz^2/4)^{1-b}\,_0F_1(2-b;-cz^2/4)\\ \end{align*} $f_2$ is not entire, so I usually just discard this solution, as I am looking to find only entire functions. What's disturbing about this, however, is that I have found a third linearly independent solution, which happens to be entire, and which is completely missed by Mathematica, and all of the references I find online: \begin{align*} f_3&:=\sum_{m=0}^\infty \frac{1}{(b)_m}\frac{(-cz^2/2)^m}{(2m)!!} \end{align*} Here, $(b)_k:=\Gamma(b+k)/\Gamma(b)$ is the Pochammer symbol. I am deeply troubled by the sight of a solution slipping past my computer algebra program: what is happening here? Is the theory behind these equations incomplete, or is there a deep explanation behind the existence of this extra solution? Why is Mathematica unable to find this holomorphic solution?