TTP01-25 Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals

Expansion of higher transcendental functions in a small parameter are needed in

   many areas of science. For certain classes of functions this can be achieved by
   algebraic means. These algebraic tools are based on nested sums and can be
   formulated as algorithms suitable for an implementation on a computer. Examples,
   such as expansions of generalized hypergeometric functions or Appell functions are
   discussed. As a further application, we give the general solution of a two-loop integral,
   the so-called C-topology, in terms of multiple nested sums. In addition, we discuss
   some important properties of nested sums, in particular we show that they satisfy a
   Hopf algebra.