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Is there a special function for linear differential equations with two regular, and one irregular singular point? I'm looking for something akin to the hypergeometric function when there are three regular points, and the confluent hypergeometric function when there is one regular singular point and one irregular singular point.

To be more concrete, I am looking at a linear ordinary differential equation of the form

\begin{equation} z(a-z)\frac{d^2u}{dz^2} + (b + cz + dz^2)\frac{du}{dz} + (f + gz)u = 0 \end{equation} where $a,b,c,d,f$, and $g$ are all constants. There are regular singular points at $z=0$ and $z=a$, and one irregular singular point at infinity. I am looking for references to solutions to equations of this type, and/or some sort of transformation that could bring this to, for example the (confluent) hypergeometric equation.

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Your example can be solved using the Heun Confluent function: according to Maple, the general solution is

$$ u \left( z \right) =c_{{1}}{{\rm e}^{zd}}{\it HeunC} \left( ad,{\frac {-a+b}{a}},{\frac {d{a}^{2}+ac+a+b}{a}},-\frac{a}{2} \left( a{d}^{2}+cd+2\, g \right) ,{\frac { \left( -2\,f+1 \right) {a}^{2}+bca+{b}^{2}}{2\;{ a}^{2}}},{\frac {z}{a}} \right) \left( z-a \right) ^{{\frac {d{a}^{2} + \left( c+1 \right) a+b}{a}}}+c_{{2}}{{\rm e}^{zd}}{\it HeunC} \left( ad,{\frac {a-b}{a}},{\frac {d{a}^{2}+ac+a+b}{a}},-\frac{a}{2} \left( a{d}^{2}+cd+2\,g \right) ,{\frac { \left( -2\,f+1 \right) {a}^{2}+bca+{b}^{2}}{2\;{a}^{2}}},{\frac {z}{a}} \right) {z}^{{ \frac {a-b}{a}}} \left( z-a \right) ^{{\frac {d{a}^{2}+ac+a+b}{a}}} $$

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  • $\begingroup$ Thanks! This makes sense in retrospect: there are four regular singular points for the Heun equation, and then two coalesce to get one irregular singular point and the confluent Heun equation. $\endgroup$ – Justin Feb 13 at 16:10

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