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?

The answer: No.

What I think that is an justification for this fact (but I'm not sure about it): It has to do with the singularity structure. For example: If we start our list with Hypergeometric functions (which is a solution to a general fuchsian equation with three regular singular points), there will be solutions to Heun's differential equation (general fuchsian equation with four regular singular points) that we cannot express. If we add Heun's functions to our list this would be the same with fuchsian equation with five regular singular points and so on. That is: no finite list as desired is possible.

My question: Is the above argument correct? If no, are there minor changes to make it correct or it is completely far from any real justification?

About this post: I had asked in meta MO about what to do. The title and the text where written thinking firstly in possible members that, as I am, also have no relevant experience on the specific topic. I had think for the title something like "About non-liouvillian special solutions to linear ODEs", but anyone who want's to know about "closed form" solutions and don't know this definition would not recognize the question as what he/she is looking for. This is also why I started with a 'simple' example and writing "closed form". I can say for myself that, four months ago, any title like this would never make me wonder if it is about what I was looking for.

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    $\begingroup$ Your answer and intuitive justification are essentially correct for LINEAR equations (the number and the nature of singularities are important, and there is no "finite list"). For non-linear equations it is completely hopeless. Linear equations can be classified in principle, into a countable number of finite-parametric families. $\endgroup$ Mar 16, 2022 at 22:49


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