  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. Another interesting example is the
Puiseux-type solution involving a differential operator generated by a
hypergeometric function of three variables. The holonomic properties of the
$F_N$ hypergeometric functions are briefly discussed.
