Some open problems related to creative telescoping

Abstract

Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still subject of ongoing research. In this paper, we present a selection of open problems in this context. We would be curious to hear about any substantial progress on any of these problems.

Constructing reductions for creative telescoping

“…Our solution to Problem 1.1 is made precise in Theorems 5.9, 3.3, and 7.1. As far as we aware of, these results are new, and provide a positive answer to [10,Problem 2.2]. The application to creative telescoping is well known; see for instance [14, section 1.2.1].…”

“…The existence of a reduction-based algorithm for general differential equations was raised as open Problem 2.2 in [10]. Problem 1.1 is essentially a more precise form of this problem, by specifying the space on which the reduction acts.…”

Computer Algebra in the Service of Enumerative Combinatorics

“…To date, there are no less than four generations of creative telescoping algorithms; examples of 4G integration algorithms are in [18] and [11]. For more details, we refer the reader to the surveys [29,32,47].…”

Telescopers for differential forms with one parameter

“…Since then, algorithms for creative telescoping have been extensively studied. Based on the techniques used in the algorithms, the existing algorithms are divided into four generations, see [13] for the details. Most recent algorithms are called reduction-based algorithms which were first introduced by Bostan et al in [6] and further developed in [7,14,15,8] etc.…”