@Carl Love Thanks a lot for the thorough answer!
But you say that certain asymptotic expansions are known? And for an arbitrary number of terms in the series? If so, I'd like to get Maple to spit out at least one of those known cases. Note that all of Maple's power-series-related commands[*1] return an arbitrary-yet-still-finite number of terms. So, please show me one simple such case from "the literature".
First, if you write the HeunC equation as
In the parameterization of Maple, in the limit z->infinity, A(z)->alpha + O(1/z), B(z)->O(1/z), then one can expect a dominating exp(-alpha*z) dependence, if boundary conditions allow it.
Regarding the literature - I noticed something similar at least in one place (eqs. 2.6-2.7). Honestly I couldn't find it in the book referenced to (Heun's Differential Equations, A. Ronveaux ed. (1995)).
There is a package for power series that's in beta testing (or perhaps it's gamma testing?), but is packaged with your stock Maple nonetheless. It's called MultiSeries. It has its own asympt command. (See ?MultiSeries.) That package has hooks for adding knowledge of new functions. Currently, it has no knowledge of HeunC (see MultiSeries:-FunctionSupported(HeunC)). Perhaps we could add sufficient knowledge under some severe restrictions on the parameters and get a useful series.
Interesting. I'll look into it, though I'm not sure how to implement "add sufficient knowledge" or put "severe restrictions".