Summation in finite terms using sage
The summation analogue of the Risch integration algorithm developed by Karr [4, 5] uses towers of difference fields to model nested indefinite sums and products, as the Risch algorithm uses towers of differential fields to model the so called elementary functions [2]. The algorithmic machinery developed by Karr, and later generalized and extended in [1, 13, 7, 6], allows one to find solutions of first order difference equations over such towers of difference fields, in turn simplifying expressions involving sums and products. We present an implementation of this machinery in the open source computer algebra system Sage [14]. Due to the nature of open source software, this allows direct experimentation with the algorithms and structures involved while taking advantage of the state of the art primitives provided by Sage. Even though these methods are used behind the scenes in the summation package Sigma [10] and they were previously implemented in [3], this is the first open source implementation.