New results for the $\epsilon$- expansion of certain one, two and three loop Feynman diagrams

For certain dimensionally-regulated one-, two- and three-loop diagrams, problems of constructing the ε -expansion and the analytic continuation of the results are studied. In some examples, an arbitrary term of the ε -expansion can be calculated. For more complicated cases, only a few higher terms in ε 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 by-product, some new relations involving Clausen function, generalized log-sine integrals and certain Euler–Zagier sums are established, and some useful results for the hypergeometric functions of argument 1 4 are presented