New results for the epsilon-expansion of certain one-, two- and three-loopFeynman diagrams

For certain dimensional ly-regulated one-, two- and three-loop diagrams, pr oblems of constructing the epsilon -expansion and the analytic continuation of the results are studied. In some examples, an arbitrary term of the eps ilon -expansion can be calculated. For more complicated cases, only a few h igher terms in epsilon are obtained. Apart from the one-loop two- and three -point diagrams, the examples include two-loop (mainly on-shell) propagator -type diagrams and three-loop vacuum diagrams. As a byproduct, some new rel ations involving Clausen function, generalized log-sine integrals and certa in Euler-Zagier sums are established, and some useful results for the hyper geometric functions of argument 1/4 are presented. (C) 2001 Elsevier Scienc e B.V. All rights reserved.