  A summation framework is developed that enhances Karr's difference field
approach. It covers not only indefinite nested sums and products in terms of
transcendental extensions, but it can treat, e.g., nested products defined over
roots of unity. The theory of the so-called $R\Pi\Sigma^*$-extensions is
supplemented by algorithms that support the construction of such difference
rings automatically and that assist in the task to tackle symbolic summation
problems. Algorithms are presented that solve parameterized telescoping
equations, and more generally parameterized first-order difference equations,
in the given difference ring. As a consequence, one obtains algorithms for the
summation paradigms of telescoping and Zeilberger's creative telescoping. With
this difference ring theory one obtains a rigorous summation machinery that has
been applied to numerous challenging problems coming, e.g., from combinatorics
and particle physics.
