Summation Theory II: Characterizations of $\boldsymbol{RΠΣ^*}$-extensions and algorithmic aspects

Abstract: Recently, $R\Pi\Sigma^*$-extensions have been introduced which extend Karr's $\Pi\Sigma^*$-fields substantially: one can represent expressions not only in terms of transcendental sums and products, but one can work also with products over primitive roots of unity. Since one can solve the parameterized telescoping problem in such rings, covering as special cases the summation paradigms of telescoping and creative telescoping, one obtains a rather flexible toolbox for symbolic summation. This article is the continuation of this work. Inspired by Singer's Galois theory of difference equations we will work out several alternative characterizations of $R\Pi\Sigma^*$-extensions: adjoining naively sums and products leads to an $R\Pi\Sigma^*$-extension iff the obtained difference ring is simple iff the ring can be embedded into the ring of sequences iff the ring can be given by the interlacing of $\Pi\Sigma^*$-extensions. From the viewpoint of applications this leads to a fully automatic machinery to represent indefinite nested sums and products in such $R\Pi\Sigma^*$-rings. In addition, we work out how the parameterized telescoping paradigm can be used to prove algebraic independence of indefinite nested sums. Furthermore, one obtains an alternative reduction tactic to solve the parameterized telescoping problem in basic $R\Pi\Sigma^*$-extensions exploiting the interlacing property.