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Expansion of higher transcendental functions in a small parameter are neededin many areas of science. For certain classes of functions this can be achievedby algebraic means. These algebraic tools are based on nested sums and can beformulated as algorithms suitable for an implementation on a computer.Examples, such as expansions of generalized hypergeometric functions or Appellfunctions are discussed. As a further application, we give the general solutionof a two-loop integral, the so-called C-topology, in terms of multiple nestedsums. In addition, we discuss some important properties of nested sums, inparticular we show that they satisfy a Hopf algebra.Comment: 28 pages, minor change

Nested sums, expansion of transcendental functions, and multiscale multiloop integrals