=ADD=
=reftype=
14
=number=
01-17
=url=
ftp://ftp.risc.uni-linz.ac.at/pub/techreports/2001/01-17.ps.gz
=year=
2001
=month=
11
=author=
Schneider; Carsten
=title=
Symbolic Summation in Difference Fields
=abstract=
There are implementations of the celebrated Gosper algorithm
(1978) on almost any computer algebra platform. Within my PhD
thesis work I implemented Karr's Summation Algorithm (1981) based
on difference field theory in the Mathematica system. Karr's
algorithm is, in a sense, the summation counterpart of Risch's
algorithm for indefinite integration. Besides Karr's algorithm
which allows us to find closed forms for a big class of multisums,
we developed new extensions to handle also definite summation
problems. More precisely we are able to apply creative telescoping
in a very general difference field setting and are capable of
solving linear recurrences in its context.

Besides this we find significant new insights in symbolic
summation by rephrasing the summation problems in the general
difference field setting. In particular, we designed algorithms
for finding appropriate difference field extensions to solve
problems in symbolic summation. For instance we deal with the
problem to find all nested sum extensions which provide us with
additional solutions for a given linear recurrence of any order.
Furthermore we find appropriate sum extensions, if they exist, to
simplify nested sums to simpler nested sum expressions. Moreover
we are able to interpret creative telescoping as a special case of
sum extensions in an indefinite summation problem. In particular
we are able to determine sum extensions, in case of existence, to
reduce the order of a recurrence for a definite summation problem.
=note=
PhD Thesis
=sponsor=
SFB grant F1305 of the Austrian "Fonds zur F\"orderung der wissenschaftlichen Forschung" and DAAD grant "Doktorandenstipendium im Rahmen des gemeinsamen Hochschulsonderprogramms III von Bund und L\"andern"
=keywords=
difference fields, symbolic summation, difference equations, d'Alembertian solutions