Can anyone please provide me with a mathematical derivation or at least some hints on how to obtain a confluent Heun equation by taking the confluence limit of the Heun equation?
We know that the Heun equation: ${\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left[{\frac {\gamma }{z}}+{\frac {\delta }{z-1}}+{\frac {\epsilon }{z-a}}\right]{\frac {dw}{dz}}+{\frac {\alpha \beta z-q}{z(z-1)(z-a)}}w=0}$, with regular singular points $0$, $1$, $a$, and $\infty$.
And the Confluent Heun eauation: ${\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left[{\frac {\gamma }{z}}+{\frac {\delta }{z-1}}+\epsilon \right]{\frac {dw}{dz}}+{\frac {\alpha z-q}{z(z-1)}}w=0}$, with regular singular points $0$ and $1$, and irregular singular point at $z = \infty$.
Now my question is: how do I obtain the confluent Heun equation by taking the confluence limit of the Heun equation?