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.
Best Answer
Your example can be solved using the Heun Confluent function: according to Maple, the general solution is
$$ u \left( z \right) =c_{{1}}{{\rm e}^{zd}}{\it HeunC} \left( ad,{\frac {-a+b}{a}},{\frac {d{a}^{2}+ac+a+b}{a}},-\frac{a}{2} \left( a{d}^{2}+cd+2\, g \right) ,{\frac { \left( -2\,f+1 \right) {a}^{2}+bca+{b}^{2}}{2\;{ a}^{2}}},{\frac {z}{a}} \right) \left( z-a \right) ^{{\frac {d{a}^{2} + \left( c+1 \right) a+b}{a}}}+c_{{2}}{{\rm e}^{zd}}{\it HeunC} \left( ad,{\frac {a-b}{a}},{\frac {d{a}^{2}+ac+a+b}{a}},-\frac{a}{2} \left( a{d}^{2}+cd+2\,g \right) ,{\frac { \left( -2\,f+1 \right) {a}^{2}+bca+{b}^{2}}{2\;{a}^{2}}},{\frac {z}{a}} \right) {z}^{{ \frac {a-b}{a}}} \left( z-a \right) ^{{\frac {d{a}^{2}+ac+a+b}{a}}} $$