# About writing solutions of linear ODE's: Is this statement correct?

A motivating example: Consider the Hypergeometric equation $$z(1-z) \frac{d^2y}{dz^2}+(c-(a+b+1)z) \frac{dy}{dz}-aby=0,$$ it has a solution given by the Gauss's Hypergeometric function $$_2F_1(a,b;c;z)=\sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n} \frac{z^n}{n!}.$$ It happens that, for general a, b and c, the hypergeometric function has not a "closed form" (I'll talk about this notion latter). That is: the Gauss's function is a special function that can express solutions to a whole class of second order ODE with polynomial coefficients.

Contextualization of the problem: About six months ago I started to think about it and if it was possible to do something similar to all solutions of second order ODE with polynomial coefficients (a particular class of Holonomic functions). So four months ago I posted THIS QUESTION on MO.

After the very useful comments from users Loïc Teyssier and Alexandre Eremenko I realized that I didn't even know how to properly ask what I had in mind. So I tried to adjust the question, but still felling that was not enough. Finally user Phil Harmsworth recommended to me the Khovanskii's paper "On solvability and unsolvability of equations in explicit form". This was a really game changer and I recommend to anyone interested in "closed form" solutions to read it.

In particular we learn that what we usually think when say "closed form" solution is called "Liouvillian solution" and is precisely defined via field extensions. After this paper the things started to get a form. Since then I read the Kovacic's "An algorithm for solving second order linear homogeneous differential equations", Bronstein's Solutions of linear ordinary differential equations in terms of special functions and this really amazing three part series (part 1, part 2 and part 3) from Paul Masson's site analyticphysics.com, wich also linked to the wonderful Slavyanov's book Special Functions: A Unified Theory Based on Singularities.

The problem: It is possible to find a finite list of non-liouvillan special functions, depending on parameters, such that all solutions to second order linear ODE with polynomial coefficients can be expressed using functions of this list and liouvillian functions?