# Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?

My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for your time and help! Here goes:

Consider the confluent hypergeometric equation

$zy''+(c-z)y'-ay=0$.

This has solutions of the form

$C_1 \Phi (a,c;z) + C_2 z^{1-c} \Phi (1+a-c,2-c;z)$,

where

$\Phi(a,c;z) = \sum_{k=0}^{\infty} \frac{(a)_k}{(c)_k} \frac{z^k}{k!}, \quad(a)_k = a(a+1)(a+2) \cdots (a+k-1)$

It is also an example of a Fuchsian differential equation.

My question is about how far the evident relation between the two linearly independent solutions to the confluent hypergeometric equation generalizes to other Fuchsian differential equations. One can think of the confluent hypergeometric equation

$zy''+ ((2-c)-z)y'- (1+a-c)y=0$

as dual in a sense to the first equation I listed, because the $\Psi$ solutions to the first equation are related to the $\Phi$ solutions of the second, and vice versa.

Because of Fuchs's Theorem guaranteeing (generically) two solutions to a Fuchsian differential equation with different asymptotic behavior for small values of the independent variable, it seems to me likely that this must be an example of some more general "duality" for a larger class of Fuchsian differential equations. Is it true that for some class of Fuchsian differential equations, one kind of solution can be regarded as "primary" (analogous to $\Phi$), and the other (the "secondary" $\Psi$ one) can be gotten in terms of the "primary" solution to another differential equation dual to the first, like for the confluent hypergeometric case?

I have a suspicion that monodromy groups might accomplish something like this, but I just have a very vague inkling of what these might be and no idea where to start reading. If someone could confirm or deny my suspicion and maybe point me to a reference, I would be very grateful!

Thanks again, and I apologize for the long post!

• I don't understand the problem? If seems to me that a linear transform of variables works perfectly well and is invertible: $$\left[\begin{array}{c} 1\\ a'\\ c' \end{array}\right]=\left[\begin{array}{ccc} 1 & 0 & 0\\ 1 & 1 & -1\\ 2 & 0 & -1 \end{array}\right]\left[\begin{array}{c} 1\\ a\\ c \end{array}\right]$$ $$\left[\begin{array}{c} 1\\ a\\ c \end{array}\right]=\left[\begin{array}{ccc} 1 & 0 & 0\\ 1 & 1 & -1\\ 2 & 0 & -1 \end{array}\right]\left[\begin{array}{c} 1\\ a'\\ c' \end{array}\right]$$ (continued next comment) – rrogers Dec 17 '15 at 17:41
• Which is the same/reflection, making the above two solutions reversed with exponent boosted to $z^{c-1}$ Which can be normalized back to the original with an overall multiplication by $z^{1-c}$. – rrogers Dec 17 '15 at 17:41
• Your observation about the relationship between the coefficients of the two confluent hypergeometric differential equations is correct, but the question is about extending this kind of relationship to other Fuchsian differential equations. Namely, for which Fuchsian differential equations does there exist a second "dual" differential equation such that the solutions of the two differential equations are related in a similar way? – Idempotent Dec 17 '15 at 18:34
• Looking back at this: I think that "Fuchsian differential equation" is of indefinite order whereas "Fuchs's Theorem" only applies to second order equations. Am I wrong? – rrogers Jan 18 '18 at 20:14