I think it's time for some clarification:
Doron Zeilberger is the one who invented the algorithm and implemented it
in Maple. He is a funny guy and let his computer Ekhad write some papers
with it...
http://www.math.rutgers.edu/~zeilberg/
There are lots of programs of this type at
http://www.math.rutgers.edu/~zeilberg/programs.html
This is maths, nothing else. As far as I know, it's not possible to
patent maths yet. (at least, I hope so) There are some extensions to the
method, as described in Marko Petkovsek, Herb Wilf, and Doron
Zeilberger's book A=B.
http://www.cis.upenn.edu/~wilf/Downld.html
(which I do not have read yet)
At the RISC in Linz, the Zeilberger METHOD was implemented for macsyma,
maxima, mma, maple and is known as "fast Zeilberger". Wegschaider
described and implemented a multisum version of "Sister Celine's technique
to find a homogeneous polynomial recurrence relation for the sum."
Again: I think you have to distinguish between maths and implementation. I
wrote to Fabricio Caruso some time ago whether he would free his maxima
implementation, but he did not respond. So I think we have to do it
ourselves...
And what about the bug???
Martin