A method for the Integration of Solutions of Ore Equations
Integration of Solutions of Linear Functional Equations
In this paper we give a simple and easy to implement method
for the following problem. Given an operator L, compute an
operator L~ of minimal order such that all solutions of L are
derivatives of solutions of L~.
If order(L~)=order(L) then we can also express each solution
of L~, i.e. the anti-derivative of a solution y of L, as a
linear combination of y and its derivatives. So in this case
we have an easy way to integrate solutions y of L.
The use of Ore rings makes our algorithm more general, so that it can
be applied to the case of difference and q-difference equations as
Note that this generalizes Gosper's algorithm in two ways,
Gosper's algorithm treats only the case when L is a difference
operator of order 1.
Furthermore our method is also easier (less technical)
than Gosper's algorithm. The technical steps are in the algorithm
for computing rational solutions, we don't need to treat these steps
because such algorithms are already given elsewhere.
For the implementation see the file integrate_sols in
Download the first version (ISSAC'97) of this paper
or pdf file.
Download the newer version (ITSF'99) of this paper:
or pdf file.