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I was wondering if the solution to the following differential equation belongs to a class of special functions. If not, is it exactly solvable?

\begin{equation} \frac{d^{2}y}{dx^{2}}+(\beta+\frac{1}{x}+\frac{\alpha}{x(x-1)})\frac{dy}{dx}-\frac{\kappa^{2}}{x(x-1)}y=0 \end{equation}

where $\beta$, $\alpha$ and $\kappa$ are real numbers.

With a change of variable I was trying to fit it in the Hypergeometric or Heun class of differential equations, but no success yet!

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  • $\begingroup$ How come? if you check the requrality of this D.E. we will find that it is irreqular. so cannot use series solution near 0 nor 1. This means the discontinuity points are non-removable so cannot have analytic solution. $\endgroup$ Commented Aug 24, 2015 at 16:52
  • $\begingroup$ Nope! $x=0$ and $x=1$ are both regular singular points. $\endgroup$
    – Ben
    Commented Aug 24, 2015 at 16:55
  • $\begingroup$ Yes you are right... sorry $\endgroup$ Commented Aug 24, 2015 at 16:58
  • $\begingroup$ This is the confluent Heun equation. $\endgroup$ Commented Aug 24, 2015 at 21:21
  • $\begingroup$ @Startwearingpurple: What form of CHE are you referring to? $\endgroup$
    – Ben
    Commented Aug 24, 2015 at 21:34

1 Answer 1

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For this to have an answer, I slightly expand my comment:

This ODE is a particular case of the confluent Heun equation $$\frac{{d}^{2}w}{{dz}^{2}}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\epsilon% \right)\frac{dw}{dz}+\frac{\alpha z-q}{z(z-1)}w=0,$$ with $\left(\alpha,\gamma,\delta,\epsilon,q\right)_{\text{Heun}}=\left(0,1-\alpha,\alpha,\beta,\kappa^2\right)_{\text{OP}}$.

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