>
>  - Generating functions. Start from an equation like 
>      f(t) = sin(t) f'(t) + 1 
>    and then get information about f (behavior at infinity ? where are 
> the poles in the complex plane ? what are the coefficients of the 
> serie expansion ? what is there growth rate ?). There is a well known 
> maple package for that called gfun which is very powerful. 
>

There are a few different things to do here:

1) generating functions themselves, series expansions: this mainly depends 
on #16137 (hint, hint!)
2) closure properties of D-algebraic, D-finite, algebraic, rational 
functions and guessing: a lot of this was recently contributed by the 
Ore-package
3) asymptotics.

Martin

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