Abstract : The mini-course is structured into three parts. In the first part, we provide a general overview of the tools available in the summation package Sigma, with a particular focus on parameterized telescoping (which includes Zeilberger's creative telescoping as a special case) and recurrence solving for the class of indefinite nested sums defined over nested products. The second part delves into the core concepts of the underlying difference ring theory, offering detailed insights into the algorithmic framework. Special attention is given to the representation of indefinite nested sums and products within the difference ring setting. As a bonus, we obtain a toolbox that facilitates the construction of summation objects whose sequences are algebraically independent of one another. In the third part, we demonstrate how this summation toolbox can be applied to tackle complex problems arising in enumerative combinatorics, number theory, and elementary particle physics.

Keywords : symbolic summation; telescoping; parameterized telescoping; recurrence solving; difference rings; difference fields