Series expansions for scattering amplitudes

This thesis investigates series expansion techniques for Feynman integrals and scattering amplitudes in quantum field theory. These integrals are reduced to linear combinations of an independent set of master integrals, via Integration By Parts identities. The master integrals, in turn, obey systems of differential equations, whose solution provides an efficient method for their evaluation. We focus on solving the differential equations using an iterative approach in the dimensional regulator, combined with a series expansion in the relevant kinematic scales. To improve convergence, we study and systematically develop Bernoulli-like variable changes, which map nearby singularities to infinity. Starting from one-scale problems, we analyze their effectiveness and limitations, identifying some of their key features. We then propose an extension of the method to multi-scale problems by introducing multiple Bernoulli-like variables. Applied to two-loop amplitudes for Higgs and Z decays into three gluons, this approach significantly reduces the number of required terms for accurate results. We also test it on a two-scale elliptic Feynman integral (the sunrise with two equal masses and a different mass), finding moderate improvements despite the complicated singularity geometry. Our results show that Bernoulli-like transformations provide a general and efficient tool for accelerating series solutions, with potential applications to high-loop, multi-scale calculations where analytic methods are intractable.