Existence Problem of Telescopers: Beyond the Bivariate Case

On the Existence of Telescopers for Rational Functions in Three Variables

Zeilberger's method of creative telescoping is crucial for the computer-generated proofs of combinatorial and special-function identities. Telescopers are linear differential or ($q$-)recurrence operators computed by algorithms for creative telescoping. For a given class of inputs, when telescopers exist and how to construct telescopers efficiently if they exist are two fundamental problems related to creative telescoping. In this paper, we solve the existence problem of telescopers for rational functions in three variables including 18 cases. We reduce the existence problem from the trivariat…

Constructing minimal telescopers for rational functions in three discrete variables

We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed by a known existence criterion of telescopers. Our approach has the important feature that it avoids the potentially costly computation of certificates. Computational experiments are also provided so as to illustrate the efficiency of our approach.

超指数函数的Hermite约化与创造性伸缩

Hermite Reduction and Creative Telescoping for Hyperexponential Functions

我们提出了一个减少算法，同时扩展了Hermite的减少有理函数和超指数函数的Hermite样减少。它产生一个唯一的加法分解，并允许决定超指数可积性。基于该约简算法，我们设计了一种计算二元超指数函数最小约简系数的新方法。它的主要特点之一是可以避免证书的昂贵计算。它的实现优于Maple的函数DEtools[Zeilberger]。此外，我们得到了一个最小的cycopers的顺序界，这是更一般和更严格的比已知的。