Gröbner Bases, Symbolic Summation and Symbolic Integration

Summary

Abstract

The treatment of combinatorial expressions and special functions by linear operators is amenable to Gröbner basis methods. In this tutorial, we illustrate the applications of Gröbner bases to symbolic summation and integration.

Introduction

In the late 1960's, Risch (1969, 1970) developed an algorithm for symbolic indefinite integration. The approach followed there consists in computing a tower of differential extensions in order to determine if an indefinite integral can be expressed in terms of elementary functions. Risch's algorithm became very popular and is now at the heart of the integration routines of many computer algebra systems. In the early 1980's, Karr (1981, 1985) appealed to similar ideas, namely difference extensions, in order to develop an algorithm for symbolic indefinite summation. Despite its indisputable algorithmic interest, Karr's algorithm has unfortunately not received as much attention as it deserves yet, due to its complexity and the difficulty to implement it.

In the early 1990's, Zeilberger (1990b) initiated a different approach to symbolic summation and integration. As opposed to the approach by differential or difference extensions, Zeilberger studies the action of algebras of differential or difference linear operators in order to compute special operators that determine the sum or integral under consideration.