Hypergeometric Functions and Feynman Diagrams

Understand

The relationship between Feynman diagrams and hypergeometric functions is discussed.

• Special attention is devoted to existing techniques for the construction of the $\epsilon$-expansion.
• As an example, we present a detailed discussion of the construction of the epsilon-expansion of the Appell function $F_3$ around rational values of parameters via an iterative solution of differential equations.
• As a by-product, we have found that the one-loop massless pentagon diagram in dimension $d=3-2\epsilon$ is not expressible in terms of multiple polylogarithms.