Differential Equations, Associators, and Recurrences for Amplitudes

We provide new methods to straightforwardly obtain compact and analytic expressions for epsilon-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different epsilon-orders of a power series solution in epsilon of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the epsilon-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also apply our tools for computing epsilon-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system). Finally, we set up our methods to systematically get compact and explicit alpha'-expansions of tree-level superstring amplitudes to any order in alpha'.