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It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably transformed into the canonical form of the Heun equation. Surely, giving this transformation in explicit terms must be possible. However, I have not found a source that actually constructs such a transformation, or perhaps provides an abstract proof of the existence of it (if that is at all possible).

I am interested in the explicit form of such a transformation - i.e. a recipe on how to obtain it for a specific given equation.

What I do know: applying a Möbius transformation to the independent variable allows to shift the $4$ regular singular points to the points $0,1,a,\infty$ where $a \in \mathbb{C}$, which is certainly necessary since these are the singular points of the canonical form of the Heun equation. Now I assume one will want to use a transformation of the dependent variable $u$ of the type $u \mapsto (x-A)^{\alpha} (x-B)^{\beta} (x-C)^{\gamma} u(x)$ or something similar. Any suggestions or references?

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The reference is

A. Ronveaux, Heun's differential equation, Oxford UP NY, 1995.

You describe the process correctly: first by a fractional linear change of the independent variable move the singular points in the position $(0,1,a,\infty)$. (One can move any 4 distinct points to such a position).

Second, by the transformation of the dependent variable (which you wrote) make one exponent at (0,1,a) equal to zero. (Your transformation with $(A,B,C)=(0,1,a)$ adds $(\alpha,\beta,\gamma)$ to all exponents at $(0,1,a)$, so this is always possible).

Now, the only equation with such regular singularities and such exponents is Heun's.

This actually proves the existence of a transformation. And gives a recipe how to perform the transformation is each particular case.

Same arguments work for a Fuchsian equation with any number of singularities which can be also brought to a standard form. This is explained in great detail, for example in Golubev, Lectures on analytic theory of differential equations (available in Russian and German), or in Ince, Ordinary differential equations (English).

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  • $\begingroup$ Thank you! I had intended to look into Ronveaux, however I could not get around the paywall protecting it. I will try to find a way though ;) $\endgroup$ – Max Jan 8 at 17:22
  • $\begingroup$ @Max: I do NOT recommend buying a Ronveaux book: it is of low value. On your question it is sufficient to use Ince or Golubev, which are really excellent. $\endgroup$ – Alexandre Eremenko Jan 8 at 18:31
  • $\begingroup$ @Alexendre: alright, thank you for the warning! $\endgroup$ – Max Jan 8 at 18:58

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