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

Abstract

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.

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

“…[12], the iterated representation for Remiddi-Vermaseren functions of complex unit was constructed. It was observed [8,10,12,52] that the physically 17 Let us recall that multiple Euler-Zagier sums are defined as ζ(s1, . .…”

“…This is just the beginning of a general analysis, but the corresponding analysis for harmonic sums is already known to be valid. [17] Unfortunately, existing computer algebra algorithms [47] do not allow us to identify the multiple series with derivatives of hypergeometric functions or their combinations. It is still matter of personal experience, but this approach looks very promising and is worthy of further analysis.…”

“…Another approach is based on extension of the algorithm of nested sums [17,18] for the study of the algebraic relations between these sums. However, there is a third approach arising from the possibility of reducing an arbitrary generalized hypergeometric function to a set of basis functions with the help of the Zeilberger-Takayama algorithm described by eq.…”

Series and ɛ-expansion of the hypergeometric functions

“…Only recently, the general algorithm for integer values of parameters has been described in [6] and its generalization has been done in [7]. The results of expansion are expressible in terms of the new functions, like harmonic polylogarithms [8] or their recent generalization [6,9]. Let us shortly describe, how this algorithm woks on the example of the generalized hypergeometric function of one variable.…”

“…They were mainly related to calculations of concrete Feynman diagrams [5]. Only recently, the general algorithm for integer values of parameters has been described in [6] and its generalization has been done in [7]. The results of expansion are expressible in terms of the new functions, like harmonic polylogarithms [8] or their recent generalization [6,9].…”