Manuel Kauers

Manuel Kauers
Johannes Kepler University, Linz, Austria

Title: Creative Telescoping via Hermite Reduction



Creative telescoping is a key tool in symbolic summation and
integration. It is used for constructing for a given definite sum or
integral an associated linear recurrence or differential equation, which
can then be used by other algorithms for finding out all sorts of
interesting facts about the quantity in question. Four generations of
creative telescoping algorithms can be distinguished: the first was
based on elimination in ideals of operator algebras. The second is the
classical Zeilberger algorithm and its variants. The third goes back to
an idea of Apagodu and Zeilberger. These algorithms are particularly
easy to implement and to analyze, but may not find optimal solutions.
The fourth and final (so far) generation of creative telescoping
algorithms is based on Hermite reduction. This idea was first worked out
for definite integrals of multivariate rational functions by Chen in his
PhD thesis. It has since been extended to more general classes of sums
and integrals. In the talk, we will explain the idea of this apprach and
a striking advantage over earlier algorithms. We will also present a
Hermite-reduction based algorithm applicable to definite hypergeometric
sums, published this year in a joint ISSAC paper with Chen, Huang, and Li.