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

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